text:
Once Upon a Time in London
Director: Simon Rumley
Stars: Terry Stone, Leo Gregory, Justin Salinger, Holly Earl, Jamie Foreman, Andy Beckwith, Dominic Keating, Geoff Bell, Kate Braithwaite, Doug Allen, Michael McKell, Roland Manookian, Joe Egan, Mark Harris, Theo Devaney, Nadia Forde, Simon Munnery
Genre: Drama, Thriller, Biopic
Review: Organised crime in London, as we know it in the form it took from the nineteen-sixties onwards, can be traced back to the rivalry between two men, Jack Comer (Terry Stone), nicknamed Spot, and Billy Hill (Leo Gregory), protégé of Spot's first big rival, Alf White (Jamie Foreman). While the Jewish gangster claimed he was the man who ensured Oswald Mosely's Blackshirts were beaten down just before World War II, he was still not a pleasant chap to be around, for the preferred method of communication among these hoodlums was violence to ensure they held onto whatever power they had. And by power, they meant money: the two went hand in hand down the years..
Simon Rumley directed this true crime yarn, one of a seeming billion low budget British crime dramas to emerge since the nineties to service the DVD and latterly, streaming market, where they could make their money back with surprising ease, there being a ready-made audience for watching your basic (at times, very basic) guns and geezers mini-epic. Here the emphasis was on gangster movies past, yes, works like The Long Good Friday, but as the title alluded, to Sergio Leone's tale of Jewish gangsters in New York Once Upon a Time in America; there were Jewish gangsters here as well, and they were just as unlovely as the ones in the Italian maestro's efforts.
This was a project whose inception came from Stone, whose life story sounded as if it would be fertile ground for precisely one of these lesser funded crime sagas (essentially, from salesman to club and dance promoter, to getting in with organised crime he did not want a part of, to reinvention in an alternate area of the entertainment world, presenting himself as an actor and filmmaker). He was not the villain of the proportions you saw played out here, but he was aware of what these people were capable of, and could at least add a sense of authenticity even if the era he had arisen during had been half a century or so after the criminals depicted in this enjoyed their heyday.
But did they enjoy it? The message here, subversive for a British gangster flick of this age, was that violence was no fun at all, curious for a genre that more often than not invited its audience to revel in actors playing yobs who thoroughly appreciated a good punch-up. The point here was that said punch-ups became the motivation rather than the accumulation of wealth, so the reputation for intimidation might have funded their lifestyles, but it was something that needed plenty of maintenance, particularly when it looked like an easy way for the powerless to gain standing. Therefore time and again Rumley staged scenes of characters getting beaten up and slashed, the perpetrators careful not to kill so they will not be subjected to the death penalty, but if you're scary enough there are perks.
Like your victims refusing to identify you to the law, so you can get away with bullying your path through life with relative impunity. However, there was nothing to savour in being king of the castle, or even a preferred underling to the king, as it was a vicious cycle you had to commit to, endlessly re-establishing yourself as a major player to the point that it meant next to nothing of any worth, you were simply the most effective thug and nobody respected you, they feared you but that was far from the same thing. Rumley filled his soundtrack with irksomely inauthentic-sounding songs for faux-period flavour, and if you had been captivated by his Crowhurst film it was unlikely you would go for this with the same enthusiasm, but there was such a tone of despair here that it was difficult for all but the most dedicated violence voyeur to get much pleasure out of it. We were watching a bunch of men and their female hangers-on chopping and punching their way through each other, with the point of their wasted lives lost on them all by the time the credits rolled and the Krays appeared. Music by Richard Chester.


text:
Q: I keep getting a syntax invalid error on my forms.py file When I run server I get syntaxerror message (invalid syntax) that indicates that the line if commit: is erroneous.
Here's my forms.py file:
from urllib import request
from django.core.files.base import ContentFile
from django.utils.text import slugify
from django import forms
from .models import Photo

from PIL import Image
from django.utils.translation import ugettext as _

class PhotoCreateForm(forms.ModelForm):

class Meta:
    model = Photo
    # fields = '__all__'
    fields = ('title', 'description', 'photo', 'license', 'location',  'url', 'slug')

    #hide url input from users with the HiddenInput widget
    widgets = {
        'url': forms.HiddenInput,
    }

def clean_title(self):
    title = self.cleaned_data.get("title")
    if len(title) > 3:
        return title
    else:
        raise forms.ValidationError("Title must be greater than 3 characters long.")

def clean_photo(self):
    photo = self.cleaned_data.get('photo', False)

    if photo:
        pho = Photo.open(photo)
        w, h = pho.size

        #validate dimensions
        max_width = max_height = 500
        if w > max_width or h > max_height:
            raise forms.ValidationError(
                _('Please use an image that is smaller or equal to '
                  '%s x %s pixels.' % (max_width, max_height)))

        #validate content type
        main, sub = photo.content_type.split('/')
        if not (main == 'photo' and sub.lower() in ['jpeg', 'pjpeg', 'png', 'jpg']):
            raise forms.ValidationError(_('Please upload a JPEG or PNG image.'))

        #validate file size
        if len(photo) > (1 * 1024 * 1024):
            raise forms.ValidationError(_('Image file too large ( maximum 1mb )'))
    else:
        raise forms.ValidationError(_("Couldn't read uploaded image"))
    return photo

def save(self, force_insert=False, force_update=False, commit=True):
    photo = super(PhotoCreateForm, self).save(commit=False)

    # generate photo name by combining photo title slug with file extension
    photo_name = '{}.{}'.format(slugify(photo.title.lower()),

    photo.photo.save(photo_name,
                     ContentFile(response.read()),
                     save=False)

    if commit:
        photo.save()
    return photo

The error suggests that the linif commit: is an invalid syntax.
I've checked the documentation and do not see this line as an invalid syntax.

redpajama_set_name: RedPajamaStackExchange


Park Bo Gum, Suzy, and Shin Dong Yup will be hosting the 'Baeksang Arts Awards' again.
The three will be hosting the 55th annual 'Baeksang Arts Awards' at the Co-ex D Hall in Gangnam, Seoul on May 1st. The trio is hosting together for the 2nd year in a row, and it's also Shin Dong Yup's 5th year hosting the show. He has hosted the 50th, 51st, 52nd, and 54th shows in the past. It's also Suzy's 4th time an MC.
The awards will air live at 9 PM KST on May 1st.


text:
Ada Health's Terms and Conditions - Covid-19 Screener
1. Welcome to Ada Covid-19 Screener
1.1. The following terms and conditions (hereinafter referred to as "Terms and Conditions") form part of the agreement between you and Ada Health GmbH (hereinafter referred to as "Ada" , "we", "our" or "us"), the owner and operator of a platform giving you both the access and the right to share our Ada COVID-19 symptom assessment and screener (hereinafter the "Covid-19 Screener"). Our Covid-19 Screener offers a question flow, adapted by country level and/or international guidance, where and when available, to help the user assess the probability that he/she may be infected with SARS-CoV-2 ("Covid-19").
1.2. The Covid-19 Screener is available to integrate into your website free of charge via a link (the "Covid-19 Link") on our website www.ada.com (hereinafter called the "Website").
PLEASE NOTE THAT THE COVID-19 SCREENER DOES NOT MAKE ANY MEDICAL DIAGNOSES. IT IS NOT A SUBSTITUTE FOR PROFESSIONAL MEDICAL ADVICE, DIAGNOSIS OR TREATMENT. THIS ASSESSMENT AIMS TO GUIDE THE USER TO THE RIGHT NEXT STEPS TO TAKE, BASED ON THE SEVERITY OF THE USER'S SYMPTOMS AND HIS/HER LOCATION. THE COVID-19 VIRUS IS A NEW DISEASE AND WE CONTINUE TO LEARN MORE EVERY DAY. IF THE USER THINKS HE/SHE MAY HAVE A LIFE-THREATENING EMERGENCY, HE/SHE SHALL IMMEDIATELY CALL HIS/HER DOCTOR OR DIAL THE EMERGENCY SERVICE LINE.
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1.6. Contract languages are German and English.
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2.1. These Terms and Conditions apply to both the copying of the Covid-19 Link and the integration of the Covid-19 Screener thus enabling you to redirect your users to our Covid-19 Screener and further contains all relevant terms and conditions related to your copying and integration of the Covid-19 Link to our Covid-19 Screener. It is vitally important that you have read and understood the present Terms and Conditions before copying and integrating the Covid-19 Link that redirects your users from your own website, or by any other available means, to our Covid-19 Screener. If you have difficulty understanding any aspect of the Terms and Conditions, please contact our legal department at legal@ada.com.
2.2. You are expressly limited to only copying the Covid-19 Link to redirect your users to our Covid-19 Screener as integrated or otherwise made available on your website if, and only if, you consent to these Terms and Conditions.
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4. The Covid-19 Screener
4.1. Please be aware that we provide the Covid-19 Screener solely for information and educational purposes only, in order to help contain the Covid-19 spread. It is not a medical consultation. The Covid-19 Screener is not a suitable substitute for medical advice obtained from the user's doctor, general practitioner or other healthcare service provider.
4.2. The Covid-19 Screener does NOT provide diagnoses for medical conditions, nor does it prescribe how the user should treat any medical issues he/she may have.
4.3. We are NOT responsible for users' individual decisions on whether to seek professional medical advice or not and further disclaim, to the greatest extent provided to us by law any liability related to such user's action or inaction. Before a user can use the Covid-19 Screener, he or she must consent to both our general Terms and Conditions as well as our Privacy Policy.
4.4. Subject to Section 8.4 of these Terms and Conditions, you understand that Ada shall have no liability to you for any claim of special, indirect, incidental, or consequential damages suffered or incurred by the users in connection with the use of the Covid-19 Screener, except in cases of malicious intent or gross negligence. You are further not entitled to initiate any legal proceedings against us or seek to compel us to join you as a defendant in a pre-existing or threatened litigation stemming from your users' use of the Covid-19 Screener.
5. Your right to integrate the Covid-19 Screener
5.1. The Covid-19 Screener and all materials and content contained therein are our property or the property of the persons who have granted us the applicable license, if any. We grant you the right to copy the Covid-19 Link and integrate the Covid-19 Screener into your website or otherwise make it available to your users, but only for the purposes of displaying the Covid-19 Screener to a larger audience. You must comply with all applicable privacy and other legal requirements in your jurisdiction at all times.
5.2. We grant you the right to: (i) copy the Covid-19 Link; and (ii) integrate (or otherwise make available) the Covid-19 Screener into your website. You are expressly prohibited from transferring these rights to anyone else, or to sell, attempt to sell or market the Covid-19 Screener for sale to anyone else. Your rights in this Section 5.2 does not preclude us from granting other persons the same or similar rights.
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5.5. Unless (i) explicitly permitted in accordance with these Terms and Conditions, or (ii) unless permitted under mandatory copyright law where necessary in copying the Covid-19 Link and integrating (or making otherwise available) the Covid-19 Screener for your and/or (iii) indispensable to obtain the information necessary to achieve the interoperability with other programs, you undertake to refrain from the following:
You agree not to replicate or copy the Covid-19 Screener, either in whole or in part,
You agree not to sell attempt to sell or market the Covid-19 Screener for sale in whole or in part to anyone else,
You agree not to modify the Covid-19 Screener, in whole or in part, in any way whatsoever.
You warrant that you will not attempt to discover or access the source code of the Covid-19 Screener, in whole or in part, unless it has been expressly published by us and released into the public domain.
5.6. You are aware that we, and/or the persons who have granted us a license in this respect, are entitled to any confidential information, rights of use, of exploitation and of recovery – and any other intellectual property rights fully or partially existing – regarding the Covid-19 Screener.
5.7. You agree that we may use your company name and trademarks, on our Website, for reference purposes only when you copy the Covid-19 Link.
6. Your integration of the Covid-19 Link to our Covid-19 Screener
6.1. In addition to the other provisions in these Terms and Conditions, this section defines certain regulations relating to the use of the Covid-19 Screener (hereinafter referred to as "Regulations").
6.2. The following is a list of activities that are prohibited in the context of integrating or making the Covid-19 Screener otherwise available to your users. You are prohibited to:
Circumvent, disable, or otherwise manipulate any of the Covid-19 Screener's security features or any features preventing or restricting the use or copying of the content that is available via the Covid-19 Screener,
Send junk mail, spam or repeated messages,
Act in any unlawful or illegal manner,
Modify, interfere with, hack or disrupt the Covid-19 Screener, or to intercept messages,
Misuse the Covid-19 Screener or to infiltrate it with viruses, trojans, worms, logic bombs or anything else that might harm the Covid-19 Screener or users using it,
Extract data from the Covid-19 Screener except as permitted under these Terms and Conditions, and
Integrate the link to our Covid-19 Screener on any abusive, threatening, obscene, misleading, untruthful, offensive or abusive website, or any other content platform that contains any form of non-consensual nudity or violence.
6.3. The non-compliance with the Regulations listed in paragraph 6.2 represents a material breach of these Terms and Conditions, and may result in us, in our sole and absolute discretion, initiating the following measures – either individually or in combination:
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7.1. Should you no longer agree to the Terms and Conditions, you can, at any time and in your sole and absolute discretion, cease sharing our Covid-19 Screener with your users.
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7.3. The Covid-19 Screener is a free tool, available to everyone. We reserve the right to restrict access to the Covid-19 Screener at any time and without notice in our sole and absolute discretion.
7.4 Either party's right to terminate the agreement for cause without notice remains unaffected. Cause that entitles us to terminate the agreement may be, in particular but not exclusively, your violation of the material contractual provisions (in particular paragraph 6.2) of these Terms and Conditions and if, due to external circumstances, it can be reasonably assumed that you are improperly integrating the Covid-19 Link and, therefore, improperly sharing our Covid-19 Screener.
8. Our offer
8.1. We grant you the right to: (i) copy the Covid-19 Link; and (ii) integrate (or otherwise make available) the Covid-19 Screener into your website, free of charge. All users visiting or otherwise accessing your website will be granted access to our Covid-19 Screener after and on the condition that they go through our onboarding flow (e.g. after they consent to both our Privacy Policy, and our general Terms and Conditions, as well as consenting to us processing their health data in order to provide them with an assessment and health advice).
8.2. Our Covid-19 Screener is not a substitute for professional medical advice, diagnosis or treatment. This assessment aims to guide users to the right next steps to take, based on the severity of their symptoms and location. The COVID-19 Coronavirus is a new disease and we continue to learn more every day. If a user thinks he/she may have a life-threatening emergency, he/she shall immediately call his/her doctor or dial the emergency service line.
8.3. Due to the nature of the internet and the given technology, we are unable to promise that users will be able to use our Covid-19 Screener at all times without any interruption and without delay or faults, and that the Covis-19 Screener will at all times meet their expectations. We can therefore assume no obligation with respect to the performance or availability of the Covid-19 Link to the Covid-19 Screener as described in these Terms and Conditions. We expressly exclude any strict liability that may be applied by enforcement of law due to defects of the Covid-19 Screener existing at the time the agreement on the use of the Covid-19 Screener between you and us was concluded.
8.4. Should you or one of the users accessing the Covid-19 Screener through your website or by any other means made available by you incur any damage through the use of the Covid-19 Screener, we may only be held liable for malicious intent and gross negligence. We may not be held liable for violation by simple negligence of any obligations other than those specified in the preceding sentences. The foregoing exclusions of liability do not apply to violations causing death or personal injury and any liability under the German Product Liability Act.
9. Third-party services
9.1. If, and to the extent that, our Covid-19 Screener contains links to other websites or resources operated by third parties, these links are to be deemed as being provided for your information and the users' information only.
9.2. We are unable to influence the content of such websites or resources provided by third parties, and we may not be held liable for the content of information or websites linked included in frames. We may not be held liable for any financial or physical damages that you or the users may incur as a result of, or in connection with, the use of such websites or resources.
10. Covid-19 Screener updates
10.1. We are constantly updating and improving the Covid-19 Screener by adapting the question flow according to international guidance and recommendations and, when and where available, national guidelines.
10.2. We reserve the right to change a specific aspect of the Covid-19 Screener, reset and suspend the service or support for that Covid-19 Screener. These changes may affect the availability of the link you incorporated into your website or by any other available means. Changes may include the removal, modification or reset of the Covid-19 Screener.
11. Changes to these Terms and Conditions
11.1. We have the right to amend, adapt or supplement the present Terms and Conditions from time to time.
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11.3. We will post a notice (hereinafter the "Notice") on our Website of any changes or additions to these Terms and Conditions at least 30 calendar days before any changes or additions to these Terms and Conditions take effect. It is your responsibility to check our website for any such Notices. If you do not object within 30 days of the Notice being posted to our Website, such changes and additions will be regarded as effective and agreed upon from the deadline onward. We will inform you of your right to object and the consequences of the objection in our notice of changes to the Terms and Conditions. If, in the event of your objection to the amendment within the agreed timeframe, it is legally, economically or technically unreasonable for us to continue servicing the Covid-19 Link supporting our Covid-19 Screener on your website, we are entitled to terminate the agreement no earlier than the effective date of the amendment as set forth in the notification. Any other right of termination remains unaffected.
12. Documents governing the contractual relationship between you and us
The current version of the Terms and Conditions includes all provisions governing the contractual relationship between you and us.
13.1. Our headquarters are located in Germany. The agreement concluded on the basis of these Terms and Conditions is subject to the laws of the Federal Republic of Germany.
13.2. The courts in Berlin – Charlottenburg, Germany have the exclusive jurisdiction to settle any disputes arising in connection with, or as a result of, your use of either the Covid-19 Link or the Covid-19 Screener.
14.1. Should you provide us with an email address, you consent to us communicating with you by electronic means (such as by email or other form of text as defined in § 126b of the German Civil Code).
14.2. The failure to enforce our rights under these Terms and Conditions is in no way a waiver of such rights.
14.3. Should any provision in these Terms and Conditions be or become void or unenforceable, the effectiveness of the remaining provisions will remain unaffected.
14.4. Only the contracting party itself has the right to enforce contractual provisions.
15.1. If you would like to contact us in regard to the present Terms and Conditions or any other document mentioned in them, please email us at legal@ada.com.

redpajama_set_name: RedPajamaCommonCrawl


Protect yourself and your furniture from sharp spindle points with one of these lovely little bowls. Added benefit is that your spindle can turn more quickly, due to reduced friction!
Sizes range from 1 1/4″ to 1 1/2″ tall and 2 1/2″ to 3″ across, with a nice dimple in the bottom of the bowl.
Made by US potters, each bowl is individually crafted.
Various colors of glazes (our choice).


Kuro, also known as Black Feather (Hopes to Stay Anonymous), is one of the original characters of the 2017 mobile game Puella Magi Madoka Magica Side Story: Magia Record.
Officially, Kuro has no summary in-game. Her first and only appearance is in the A la Carte Valentine event.
Kuro has no side story. Her side story chapters are titled "Black Magical Girl Quest 1-3" and feature no cutscenes of any kind, instead simply having standard battles with a variety of enemies. When selecting the individual battle, Kuro stands to one side in front of different backgrounds, such as Magius' hideout or a classroom and saying nothing.
"Matasaburo" is a reference to Kenji Miyazawa's work 'Kaze no Matasaburo' about a child who disappears as soon as they came.
Her illustrator is unknown. Her card and transformation sequence use a Live 2D model, her character designer is Sasagi Koushi, and her story was written by Gekidan Inu Curry.
Homura's Valentine's story was made available on February 8th. Kuro was released on the 15th of that month. Since she was only added due to fan response, Kuro must have been thrown together in a week.
Kuro is the second 1* after Iroha Tamaki, and the only magical girl to be capped at 1*.
Kuro is also the first magical girl originally without a unique memoria. Her unique memoria was later provided as an event shop item the following year in the 2019 Valentine's Day event.
Attack UP Upon Death [IV] (Allies / 2 T) & Defense UP Upon Death [IV] (Allies 2 T) Passive The chocolate I had wanted to give you is here.
Thinking you would be pleased, I had prepared the chocolate. But, I'm sorry, I can no longer give it to you. I'll no longer be myself, and this chocolate will surely taste bitter.


text:
\section{Introduction}

\noindent
A \emph{maximum distance separable} (MDS) code $\mathcal{C}(n,k,d)$ of length $n$, dimension $k$, and minimum distance $d$ is a linear code attaining the Singleton bound, i.e., $d=n-k+1$ \cite{singleton1964maximum}.
The most prominent MDS codes are \emph{generalized Reed--Solomon} (GRS) codes \cite{reed1960polynomial}.
However, there are many other known constructions for MDS codes, e.g., based on the equivalent problem of finding $n$-arcs in projective geometry \cite{macwilliams1977theory}, circulant matrices \cite{roth_construction_1989}, Hankel matrices \cite{roth1985generator}, or extending GRS codes.

Recently, Sheekey \cite{sheekey_new_2015} introduced a new class of \emph{maximum rank distance} codes---which are MDS in terms of the \emph{rank metric}.
These codes, \emph{Twisted Gabidulin} codes, were shown to be not equivalent to Gabidulin codes (the rank-metric analogues of Reed--Solomon codes).

In this paper, we introduce a new construction of Hamming-metric MDS codes, inspired by~\cite{sheekey_new_2015}.
The idea is to evaluate polynomials of the form
\[
  f(x) = a_0 + a_1 x + \ldots + a_{k-1} x^{k-1} + \eta a_h x^{k-1+t} ,
\]
for some $0 \leq h < k$ and $t < n-k$.
By a prudent choice of $t, h$ and $\eta$, we ensure that any such polynomial has at most $k-1$ zeroes among the evaluation points, even though their degree is larger than $k-1$.
This is enough to ensure that the resulting code is MDS.

We single out two explicit subclasses of this construction where the MDS property can be a priori ensured.
These contain codes of length up to roughly $q/2$ for any $q$, and we show that for $q \geq 11$, they contain non-GRS MDS codes.

The results we obtain are somewhat reminiscent of results in \cite{roth_construction_1989}, where non-GRS MDS codes were constructed of length roughly $q/2$ for even $q$.
However, our construction is very different, and for small values of $q$ we verified using a computer that our construction produces codes inequivalent to the ones mentioned in \cite{roth_construction_1989}.
More importantly, our construction also gives a very simple way to produce non-GRS MDS codes of length at least $q/2$ if $q$ is odd and $q \ge 11$.

Besides adding new codes to the family of known MDS codes, the new code class might be interesting for code-based cryptography.
As future work, we will analyze whether our codes or their subfield subcodes are suitable for this purpose.

\section{Preliminaries}

\noindent
In this section, we recall several definitions and known results for future use in the paper. We denote by $\mathbb{F}_q$ the finite field with $q$ elements.

\begin{definition}[\!\!\cite{roth2006introduction}]
Let $\alpha_1,\dots,\alpha_n \in \mathbb{F}_q \cup \{\infty\}$ be distinct, $k<n$, and $v_1,\dots,v_n \in \mathbb{F}_q^*$. The corresponding \emph{generalized Reed--Solomon (GRS) code} is defined by
\begin{align*}
\CGRS{n,k} = \left\{ [v_1 f(\alpha_1), \dots, v_n f(\alpha_n)] : \, f \in \mathbb{F}_q[x], \, \deg f <k \right\}.
\end{align*}
In this setting, for a polynomial $f$ of degree $\deg f < k$, the quantity $f(\infty)$ is defined as $a_{k-1}$, the coefficient of $x^{k-1}$ in the polynomial $f$.
\end{definition}

In case $v_i=1$ for all $i$, the code is called a \emph{Reed--Solomon (RS) code}. Any non-zero evaluation polynomial $f$ is of degree $\deg f <k$ and hence has at most $k-1$ roots among the evaluation points $\alpha_1,\dots,\alpha_n$ in $\mathbb{F}_q$. If $f(\infty)$ ``evaluates'' to zero, this just means that $\deg f < k-1$ and hence $f$ has at most $k-2$ roots among the remaining evaluation points. This proves that a GRS code is MDS.

In this article, we will construct MDS codes of length $n$ and dimension $k$ using spaces of polynomials that may contain elements of degree $\deg f \geq k$. To define our codes, we will use the following map.

\begin{definition}\label{def:eval_map}
Let $\mathcal{V} \subset \mathbb{F}_q[X]$ be a $k$-dimensional $\mathbb{F}_q$-linear subspace. Let $\alpha_1,\dots,\alpha_n \in \mathbb{F}_q \cup \{\infty\}$ be distinct and write ${\ve \alpha} = [\alpha_1,\dots,\alpha_n]$. We call $\alpha_1,\dots,\alpha_n$ the \emph{evaluation points}. Then we define the \emph{evaluation map} of ${\ve \alpha}$ on $\mathcal{V}$ by
\begin{align*}
\ev{\cdot}{{\ve \alpha}} : \mathcal{V} \to \mathbb{F}_q^n, \quad
f \mapsto [f(\alpha_1),\dots,f(\alpha_n)].
\end{align*}
Here $f(\infty)$ is defined as $a_{\ell}$, the coefficient of $x^{\ell}$ in the polynomial $f \in \mathcal{V}$, where $\ell:=\max \deg\{f : f \in \mathcal{V}\}.$
\end{definition}

The evaluation map above is $\mathbb{F}_q$-linear. This means in particular that if for a given $\mathcal{V}$ the evaluation map is injective, then the code $\ev{\mathcal V}{{\ve \alpha}} \subset \mathbb{F}_q^n$ will be an $\mathbb{F}_q$-linear code of length $n$ and dimension $k$.
Injectivity is immediate if $\ell < n$.
If $\mathcal{V}$ consists of all polynomials of degree strictly less than $k$, the resulting code is an RS code.
For other choices of $\mathcal{V}$, the resulting code might still be \emph{equivalent} to an RS code; we use the following notion of code equivalence.

\begin{definition}
Let $\mathcal{C}_1,\mathcal{C}_2$ be $\mathbb{F}_q$-linear $[n,k]$ codes. We say that $\mathcal{C}_1$ and $\mathcal{C}_2$ are \emph{equivalent} if there is a permutation $\pi \in S_n$ and $\v := [v_1,\dots,v_n] \in \left(\mathbb{F}_q^*\right)^n$ such that $\mathcal{C}_2 = \varphi_{\pi,\v}\left( \mathcal{C}_1 \right)$ where $\varphi_{\pi, \v}$ is the Hamming-metric isometry
\begin{align*}
\varphi_{\pi,\v} : \mathbb{F}_q^n \to \mathbb{F}_q^n, \quad
[c_1,\dots,c_n] \mapsto [v_1 c_{\pi(1)},\dots,v_n c_{\pi(n)}].
\end{align*}
\end{definition}

It is clear from the above definition that a code is a GRS code if and only if it is equivalent to an RS code.
The following well-known theorem provides an effective tool to decide whether a code is equivalent to an RS code.

\begin{theorem}[\!\!\cite{roth1985generator,roth1989mds}]\label{thm:GRS_characterization}
A linear code with generator matrix $\ve G = [\ve I \mid \ve A]$ is a GRS code if and only if
\begin{enumerate}[label=(\roman*)]
\item\label{itm:GRS_characterization_1} All entries of $\ve A$ are non-zero.
\item\label{itm:GRS_characterization_2} All $2 \times 2$ minors of $\ve{\tilde{A}}$ are non-zero, and
\item\label{itm:GRS_characterization_3} All $3 \times 3$ minors of $\ve{\tilde{A}}$ are zero,
\end{enumerate}
where $\ve{\tilde{A}} \in \mathbb{F}_q^{k \times n-k}$ is given by $\tilde{A}_{ij} = A_{ij}^{-1}$.
\end{theorem}

Note that any MDS code has a generator matrix of the form $\ve G = [\ve I \mid \ve A]$ and that items \ref{itm:GRS_characterization_1} and \ref{itm:GRS_characterization_2} above are satisfied for this $A$. Hence the difference between GRS and non-GRS MDS codes will only become apparent using item \ref{itm:GRS_characterization_3}. If $k<3$ or $n-k<3$, the matrix $A$ has no $3 \times 3$ minors, so the following corollary holds.
\begin{corollary}\label{cor:min_k_n-k_>_2}
Suppose that $k<3$ or $n-k<3$. Any MDS code of length $n$ and dimension $k$ is equivalent to an RS code.
\end{corollary}

We finish this section by quoting results on $t$-sum generators in abelian groups from \cite{roth_construction_1989, roth_t-sum_1992}. We will apply these results to the abelian groups $(\mathbb{F}_q^*,\cdot)$ and $(\mathbb{F}_q,+)$ to analyse several instances of our code construction in the coming sections.

\begin{definition}[\!\!\cite{roth_t-sum_1992}]
Let $(A,\oplus)$ be a finite abelian group and $k \in \mathbb{N}$. A subset $S \subset A$ is called a \emph{$k$-sum generator} of $A$ if for all $a \in A$, there are distinct $s_1,\dots,s_k \in S$ such that $a=\bigoplus_{i=1}^{k} s_i$. We denote by $M(k,A)$ the smallest integer such that any $S \subset A$ with $|S|>M(k,A)$ is a $k$-sum generator of $A$.
\end{definition}

\begin{lemma}[\!\!{\cite[Theorem~3.1]{roth_t-sum_1992}}]\label{lem:tsum_roth_theorem_even_order}
Let $|A|= 2r$ for some $r \geq 6$. Then, for any $3 \leq k \leq r-2$,
\begin{align*}
M(k,A) = r,
\end{align*}
except if $A \in \{\mathbb{Z}_2^m,\mathbb{Z}_4 \times \mathbb{Z}_2^{m-1}\}$ for some $m > 1$ and $k \in \{3,r-2\}$ in which case
\begin{align*}
M(k,A) = r+1.
\end{align*}
\end{lemma}
This lemma suffices for our purposes. See \cite{roth_construction_1989, roth_t-sum_1992} for more information in case $|A|$ is odd.

\section{Twisted Reed--Solomon Codes}\label{sec:CTRS_one_twist}

\noindent
In this section, we present our new code construction.
Similar to RS codes, we evaluate polynomials whose first $k$ coefficients we can choose arbitrarily.
The difference is that we allow another monomial of degree larger than $k-1$ to occur in the polynomials as well.
We define a set of evaluation polynomials as follows.

\begin{definition}\label{def:twisted_polynomials}
Let $k,t,h \in \mathbb{N}$ such that $0 \le h<k\leq q$ and let $\eta \in \mathbb{F}_q \backslash \{0\}$. Then, we define the set of \emph{$(k,t,h,\eta)$-twisted  polynomials} by
\begin{align*}
\mathcal{V}_{k,t,h,\eta} = \left\{ f = \sum_{i=0}^{k-1} a_i x^i + \eta a_h x^{k-1+t} : a_i \in \mathbb{F}_q \right\},
\end{align*}
where we call $h$ the \emph{hook} and $t$ the \emph{twist}.
\end{definition}

Note that $\mathcal{V}_{k,t,h,\eta} \subseteq \mathbb{F}_q^n$ is a $k$-dimensional $\mathbb{F}_q$-linear subspace. Using the evaluation map from Definition \ref{def:eval_map} for $\mathcal{V}=\mathcal{V}_{k,t,h,\eta}$, we obtain the codes that we will study in this article.

\begin{definition}\label{def:CTRS_one_twist}
Let $\alpha_1,\dots,\alpha_n \in \mathbb{F}_q \cup \{\infty\}$ be distinct and write ${\ve \alpha} = [\alpha_1,\dots,\alpha_n]$.
Let $k,t,h,\eta$ be chosen as in Definition~\ref{def:twisted_polynomials} such that $k<n$ and $t \leq n-k$. Then, the corresponding \emph{twisted Reed--Solomon code} of length $n$ and dimension $k$ is given by
\begin{align*}
\TRS{{\ve \alpha},t,h,\eta}{k}{} = \ev{\mathcal{V}_{k,t,h,\eta}}{{\ve \alpha}} \subseteq \mathbb{F}_q^n.
\end{align*}
\end{definition}

For brevity, we will use the phrase \emph{twisted codes} rather than \emph{twisted Reed--Solomon codes} from now on.
Note that $\TRS{{\ve \alpha},t,h,\eta}{k}{}$ indeed has dimension $k$ since the evaluation map is injective: any polynomial $f \in \mathcal{V}_{k,t,h,\eta}$ satisfies $\deg f \le k-1+t < n$. In principle $\eta$ could be $0$ in the above definition, but in that case we simply obtain RS codes.

\section{MDS Twisted Codes}
\label{sec:MDS_TRS_Codes}

In general, twisted codes are not MDS for all parameters ${\ve \alpha},k,t,h,\eta$. However, in this section we will describe several classes of twisted codes that are always MDS.

\subsection{$(*)$-twisted  Codes}
\label{ssec:Max_Length_MDS_Primitively Twisted}

\noindent
If $(t,h)=(1,0)$, it is possible to give a succinct condition on when the code $\TRS{1,0,\eta,{\ve \alpha}}{k}{}$ is MDS. More precisely, we have the following:
\begin{lemma}\label{lem:cond_MDS_very_simple_twist}
Let $k<n$, $\alpha_1,\dots,\alpha_n \in \mathbb{F}_q$ distinct and $\eta \in \mathbb{F}_q$. Then the twisted code $\TRS{{\ve \alpha},1,0,\eta}{k}{}$ is MDS if and only if
\begin{align}
\eta (-1)^k \prod_{i \in \mathcal{I}} \alpha_i \neq 1 \quad \forall \mathcal{I} \subseteq \{1,\dots,n\} \textrm{ s.t. } |\mathcal{I}|=k. \label{eq:cond_MDS_very_simple_twist}
\end{align}
\end{lemma}

\begin{proof}
The code is MDS if and only if the only polynomial of the required form which has $k$ roots among the $\alpha_i$ is the zero polynomial. Let $f = \sum_{i=0}^{k-1} a_i x^i + \eta a_0 x^k$ be such a polynomial. In the first place $a_0 \neq 0$, since otherwise $\deg f < k$, making it impossible that $f$ has $k$ roots among the $\alpha_i$. If there is a subset $\mathcal{I} \subseteq \{1,\dots,n\}$ with $|\mathcal{I}|=k$ and $f(\alpha_i) = 0$ for all $i \in \mathcal{I}$, we can write $f = \eta a_0 \prod_{i \in \mathcal{I}}(x-\alpha_i)$ and considering the constant term it follows that
\begin{align}
1 = \eta (-1)^k \prod_{i \in \mathcal{I}} \alpha_i. \label{eq:intermediate_step_MDS_very_simple_twist}
\end{align}
If Condition \eqref{eq:cond_MDS_very_simple_twist} is satisfied, no such $f$ can exist.
Conversely, if there is an $\mathcal{I}$ such that $\eta (-1)^k \prod_{i \in \mathcal{I}} \alpha_i = 1$, then $\eta\neq 0$ and $f = \eta \prod_{i \in \mathcal{I}}(x-\alpha_i) \in \mathcal{V}_{k,t,h,\eta}$ has $k$ roots among the $\alpha_i$, so the code is not MDS.
\end{proof}

This leads to our first explicit subclass of twisted codes which are MDS\footnote{%
  This class can be seen as the Hamming-metric analog of Twisted Gabidulin codes \cite{sheekey_new_2015}. However, we use different techniques for analyzing our codes.}:
\begin{definition}\label{def:startwisted}
  $\TRS{{\ve \alpha},1,0,\eta}{k}{}$ is a \emph{$(*)$-twisted  code} if the elements of ${\ve \alpha}$ are a subset of $G \cup \{ 0 \}$, for $G$ a proper subgroup of $(\mathbb{F}_q^*, \cdot)$, and if $(-1)^k \eta^{-1} \in \mathbb{F}_q^* \setminus G$.
  We write $\TRS{{\ve \alpha},\eta}{k}{*} := \TRS{{\ve \alpha},1,0,\eta}{k}{}$.
\end{definition}

\begin{theorem}\label{thm:MDS_property_simple_twist}
Any $(*)$-twisted  code is an MDS code.
\end{theorem}
\begin{proof}
If a $(*)$-twisted  code $\TRS{{\ve \alpha},\eta}{k}{*}$ is not MDS, then Lemma \ref{lem:cond_MDS_very_simple_twist} implies that there exists $\mathcal{I} \subseteq \{1,\dots,n\}$ such that $(-1)^k\eta\prod_{i \in \mathcal{I}} \alpha_i =1$. Since the $\alpha_i$ are contained in a subgroup $G$ of $\mathbb{F}_q^*$, we have $\prod_{i \in \mathcal{I}} \alpha_i \in G$, implying that $(-1)^k\eta \in G$ as well. This gives a contradiction.
\end{proof}

\begin{corollary}\label{cor:MDS_primitively-twisted}
Let $\mathbb{F}_q$ be a finite field and let $p$ be a prime divisor of $q-1$. Then there exists a $(*)$-twisted  code of length
\begin{align*}
n = \tfrac{q-1}{p}+1.
\end{align*}
In particular, if $q$ is odd, $(*)$-twisted  codes can have length $n = \frac{q+1}{2}$.
\end{corollary}
\begin{proof}
The maximum cardinality of a proper subgroup $G$ of $\mathbb{F}_q^*$ is $(q-1)/p$. Now let ${\ve \alpha}$ be $G \cup \{0\}$ in some order in Definition \ref{def:startwisted}.
\end{proof}

For odd $q$ $(*)$-twisted  codes can therefore be rather long.
But the choice of ${\ve \alpha}$ is very limited in \cref{def:startwisted}, and perhaps longer codes could be constructed with $(t,h) = (1,0)$.
The following answers this negatively, by using $k$-sum generators.
\begin{lemma}\label{lem:k_sum_gen_star_MDS}
Let $\eta \in \mathbb{F}_q^*$ and let $S := \{\alpha_1,\dots,\alpha_n\} \subseteq \mathbb{F}_q^*$ be a $k$-sum generator of $(\mathbb{F}_q^*,\cdot )$.
Then the twisted code $\TRS{{\ve \alpha},1,0,\eta}{k}{}$ is not MDS.
\end{lemma}
\begin{proof}
By the definition of a $k$-sum generator, there is an index set $\mathcal{I} \subseteq \{1,\dots,n\}$ with $|\mathcal{I}| = k$ such that $\prod_{i \in \mathcal{I}} \alpha_i = (-1)^k \eta^{-1}$.
Hence $\TRS{{\ve \alpha},1,0,\eta}{k}{}$ is not MDS by \cref{lem:cond_MDS_very_simple_twist}.
\end{proof}

\begin{theorem}\label{thm:inverse_mds_statement_very_simple_twist}
Let $\mathbb{F}_q$ be a finite field, with $q$ odd. Further let $\eta \in \mathbb{F}_q^*$. Then, for any $3 \leq k \leq \tfrac{q-1}{2}-2$, the length $n$ of a twisted code $\TRS{{\ve \alpha},1,0,\eta}{k}{}$ which is MDS, satisfies
$
n \leq \frac{q+1}{2}.
$
\end{theorem}

\begin{proof}
We know that $(\mathbb{F}_q^*,\cdot)$ is a cyclic abelian group of order $|\mathbb{F}_q^*|=q-1$, which is even since $q$ is a power of an odd prime. Hence \cref{lem:tsum_roth_theorem_even_order} implies
\begin{align*}
M(k,\mathbb{F}_q^*) = \tfrac{q-1}{2}.
\end{align*}
Now suppose $n>\tfrac{q+1}{2}$ and ${\ve \alpha}=[\alpha_1,\dots,\alpha_n]$. Then, $S := \{\alpha_1,\dots,\alpha_n\} \cap \mathbb{F}_q^*$ has cardinality
\begin{align*}
|S| \geq n-1 > \tfrac{q+1}{2}-1 = \tfrac{q-1}{2} = M(k,\mathbb{F}_q^*)
\end{align*}
and is therefore a $k$-sum generator of $\mathbb{F}_q^*$.
By \cref{lem:k_sum_gen_star_MDS}, the code $\TRS{{\ve \alpha},1,0,\eta}{k}{}$ is not MDS for any $\eta \neq 0$.
\end{proof}

\begin{remark}
For even $q>4$, the $(*)$-twisted  codes cannot attain length $\lfloor(q+1)/2\rfloor$. However, we determined by computer search that for e.g.~$q=16$, there are many $[9,k]$ twisted codes with $(t,h)=(1,0)$ for other choices of ${\ve \alpha}$ and $\eta$, for $k = 3,4,5$.
See also \cref{sec:Computer_Searches}.
\end{remark}

\subsection{$(+)$-twisted  Codes}
\label{ssec:Max_Length_MDS_Additive-Primitively_Twisted}

\noindent
While the results in the previous subsection were based on properties of the multiplicative group $(\mathbb{F}_q^*,\cdot)$, it is also possible to use the structure of the additive group $(\mathbb{F}_q,+)$. This structure arises when considering the case $(t,h)=(1,k-1)$.
We have the following analogue of \cref{lem:cond_MDS_very_simple_twist}.
\begin{lemma}\label{lem:cond_MDS_primitively_twisted_h=k-1}
Let $k<n\leq q$, $\alpha_1,\dots,\alpha_n \in \mathbb{F}_q$ distinct and $\eta \in \mathbb{F}_q$. Then the twisted code $\TRS{{\ve \alpha},1,k-1,\eta}{k}{}$ is MDS if and only if
\begin{align}
\eta \sum_{i \in \mathcal{I}} \alpha_i \neq -1 \quad \forall \mathcal{I} \subseteq \{1,\dots,n\} \textrm{ s.t. } |\mathcal{I}|=k. \label{eq:cond_MDS_primitively_twisted_h=k-1}
\end{align}
\end{lemma}
\begin{proof}
  The code is MDS if and only if the only polynomial
  \[
    f = a_0 + \ldots + a_{k-1}x^{k-1} + \eta a_{k-1} x^k \in \mathcal{V}_{k,1,k-1,\eta}
  \]
  having $k$ roots among the $\alpha_i$ is the zero polynomial.
If $f \neq 0$ is such a polynomial, there is a subset $\mathcal{I} \subseteq \{1,\dots,n\}$ with $|\mathcal{I}|=k$ and $f(\alpha_i) = 0$ for all $i \in \mathcal{I}$.
Writing $f = \eta a_{k-1} \prod_{i \in \mathcal{I}}(x-\alpha_i)$, with $a_{k-1} \neq 0$, we obtain a contradiction by considering the coefficient of $x^{k-1}$ on both sides, since
\begin{align*}
  a_{k-1} = -\eta a_{k-1} \sum_{i \in \mathcal{I}} \alpha_i
  \iff
  \eta \sum_{i \in \mathcal{I}} \alpha_i = -1
\end{align*}
Conversely, if there is an $\mathcal{I}$ such that $\eta \sum_{i \in \mathcal{I}}(-\alpha_i) = 1$, then $f = \eta \prod_{i \in \mathcal{I}}(x-\alpha_i)$ is a polynomial of the appropriate form having $k$ roots among the $\alpha_i$, so the code is not MDS.
\end{proof}

As in the previous subsection, this naturally gives rise to a subclass of twisted codes.
\begin{definition}\label{def:plustwisted}
  $\TRS{{\ve \alpha},1,0,\eta}{k}{}$ is a \emph{$(+)$-twisted  code} if the elements of ${\ve \alpha}$ are a subset of $V \cup \{\infty\}$, for $V$ a proper subgroup of $(\mathbb{F}_q, +)$, and if $\eta^{-1} \in \mathbb{F}_q\setminus V$.
  We write $\TRS{\eta,\ast}{k}{+} := \TRS{{\ve \alpha},1,0,\eta}{k}{}$.
\end{definition}

The analysis of these codes follows that of their multiplicative counterparts from the previous subsection very closely.
In particular we have the following:
\begin{theorem}\label{thm:MDS_property_primitively_twisted_h=k-1}
Any $(+)$-twisted  code is an MDS code.
\end{theorem}
\begin{proof}
We simply apply \cref{lem:cond_MDS_primitively_twisted_h=k-1} instead of Lemma \ref{lem:cond_MDS_very_simple_twist}. Some care must be taken that adding $\infty$ to a set of evaluation points preserves the MDS property. However, if a polynomial $f \in \mathcal{V}_{k,1,k-1,\eta}$ satisfies $f(\infty)=0$, that means its degree is at most $k-2$.
\end{proof}

\begin{corollary}\label{cor:MDS_primitively-twisted_h=k-1}
Let $\mathbb{F}_q$ be a finite field of characteristic $p$. Then there exists a $(+)$-twisted  code of length
\begin{align*}
n = \tfrac{q}{p}+1.
\end{align*}
In particular, if $q$ is even, $(+)$-twisted  codes can have length $n = \frac{q}{2}+1$.
\end{corollary}
\begin{proof}
The maximum cardinality of a proper subgroup $V$ of $\mathbb{F}_q$ is $q/p$. Now set $S=\{\infty\} \cup V$ in Definition \ref{def:plustwisted}. \end{proof}

Lemma \ref{lem:k_sum_gen_star_MDS} and Theorem \ref{thm:inverse_mds_statement_very_simple_twist} have a direct analogue as well. For completeness, we state the results, but since the proofs are extremely similar, we leave these to the reader.
\begin{lemma}
  \label{lem:k_sum_gen_plus_MDS}
  Let $\eta \neq 0$ and $S := \{\alpha_1,\dots,\alpha_n\} \in \mathbb{F}_q$ be a $k$-sum generator of $(\mathbb{F}_q,+)$.
  Then the twisted code $\TRS{{\ve \alpha},1,k-1,\eta}{k}{}$ is not MDS.
\end{lemma}

\begin{theorem}\label{thm:inverse_mds_statement_very_simple_twist_h=k-1}
Let $\mathbb{F}_q$ be a finite field, with $q$ even. Further let $\eta \in \mathbb{F}_q^*$. Then, for any $3 \leq k \leq \tfrac{q}{2}-2$, the length $n$ of a twisted code $\TRS{{\ve \alpha},1,k-1,\eta}{k}{}$ which is MDS, satisfies $n \leq \frac{q}{2}+1$ if $3< k < \frac{q}{2}-3$ and $n \leq \frac{q}{2}+2$ if $k \in \{3,\frac{q}{2}-2\}.$
\end{theorem}

\subsection{General Theory of Twisted Codes}
\label{subsec:Max_Length_MDS_General}

For general $t$ and $h$, it is still possible to derive a criterion for a code $\TRS{{\ve \alpha},t,h,\eta}{k}{}$ to be MDS. We do so in the following lemma, generalizing Lemmas \ref{lem:cond_MDS_very_simple_twist} and \ref{lem:cond_MDS_primitively_twisted_h=k-1}.

\begin{lemma}\label{lem:MDS_property_one_twist}
Let $\alpha_1,\dots,\alpha_n \in \mathbb{F}_q$ and define for $\mathcal{I} \subseteq \{1,\dots,n\}$ the polynomial $\sum_i \sigma_i x^i := \prod_{i \in \mathcal{I}} (x-\alpha_i)$, where $\sigma_i := 0$ for $i<0$. The twisted code $\TRS{{\ve \alpha},t,h,\eta}{k}{}$ is MDS if and only if the matrix
\begin{align*}
\underset{=: \, \ve A_\mathcal{I} \, \in \mathbb{F}_q^{t \times t}}{\underbrace{
\begin{bmatrix}
\eta^{-1} -\sigma_{h-t+1} & -\sigma_{h-t+2} & \dots  & -\sigma_{h} & \\
\sigma_{k-1} & 1 & & & & \\
\sigma_{k-2} & \sigma_{k-1} & 1 &  & & \\
\sigma_{k-3} & \sigma_{k-2} & \sigma_{k-1} & 1 &  & & \\
\vdots & \ddots & \ddots & \ddots & \ddots & & \\
\sigma_{k-t+1} & \dots &\sigma_{k-3} & \sigma_{k-2} & \sigma_{k-1} & 1\\
\end{bmatrix}
}},
\end{align*}
is regular for all $\mathcal{I} \subseteq \{1,\dots,n\}$ such that $|\mathcal{I}|=k$.
\end{lemma}

\begin{proof}
Let $f \in \mathcal{V}_{k,t,h,\eta}$ be a polynomial with at least $k$ roots among the $\alpha_i$'s.
Then, there is an index set $\Iset \subseteq \{1,\dots,n\}$ with $|\Iset|=k$ and $f(\alpha_i) = 0$ for all $i \in \Iset$.
We can factor $f$ into $f(x) = g(x) \cdot \sigma(x)$, with
\begin{align*}
g(x) = \sum\limits_{i=0}^{t-1} g_i x^i, \quad
\sigma(x) = \sum\limits_{i=0}^{k} \sigma_i x^i := \prod_{i \in \Iset} (x-\alpha_i).
\end{align*}
Note that $\sigma_k=1$.
Since by construction, the coefficients in $f$ to $x^k, \ldots, x^{k+t-2}$ are zero, we obtain the following system of $(t-1)$ equations in the $g_j$'s,
\begin{align}
0 = \sum\limits_{j=0}^{i} g_j \sigma_{i-j} \quad i=k,\dots,k+t-2, \label{eq:one_hook_MDS_first_equations}
\end{align}
where $g_j := 0$ for all $j \notin \{ 0,\ldots, t-1 \}$, and $\sigma_j:=0$ for $j \notin \{0,\ldots,k\}$.
Considering the coefficients of $x^{k-1+t}$ and $x^h$, we obtain
$\eta a_h   = g_{t-1}$ and $a_h  = \sum_{j=0}^{h} g_j \sigma_{h-j}$
and hence
\begin{align}
0 = \eta^{-1} g_{t-1} -\sum\limits_{j=0}^{h} g_j \sigma_{h-j}. \label{eq:one_hook_MDS_fourth_equation}
\end{align}
Equations \eqref{eq:one_hook_MDS_first_equations} and \eqref{eq:one_hook_MDS_fourth_equation} result in a homogeneous system of $t$ equations in $t$ variables $\ve g := [g_{t-1},g_{t-2},\dots,g_{0}]^\mathrm{T}$:
\begin{align}
\ve A_\mathcal{I} \cdot \ve g
=
\ve 0 \label{eq:MDS_proof_system_one_twist}
\end{align}
The code $\TRS{{\ve \alpha},t,h,\eta}{k}{}$ is MDS if and only if the only polynomial $f \in \mathcal{V}_{k,t,h,\eta}$ with at least $k$ roots is the zero polynomial, which holds if and only if the system \eqref{eq:MDS_proof_system_one_twist} has only the zero vector as solution for all choices of index sets $\mathcal{I} \subseteq \{1,\dots,n\}$ with $|\mathcal{I}|=k$. This implies the claim.
\end{proof}

\begin{remark}
If one includes $\infty$ as evaluation point and $h=k-1$, the above lemma is still true when considering $\mathcal{I}$ not containing $\infty$. Indeed if $f(\infty)=0$, then $\deg f < k-1$, which means the MDS property is not affected.
\end{remark}

As we will see in \cref{sec:Computer_Searches} many long MDS codes can be obtained using twisted codes for particular values of $q$. Hence an upper bound like in Theorems \ref{thm:inverse_mds_statement_very_simple_twist} and \ref{thm:inverse_mds_statement_very_simple_twist_h=k-1} does not hold for general $h$ and $t$.
On the other hand, it seems harder to find explicit constructions of such long MDS codes.
We do have the following result.
\begin{theorem}\label{thm:MDS_property_one_twist}
Let $\mathbb{F}_s \subsetneq \mathbb{F}_q$ be a proper subfield and $\alpha_1,\dots,\alpha_n \in \mathbb{F}_s$.
If $\eta \in \mathbb{F}_q \setminus \mathbb{F}_s$, then the twisted code $\TRS{{\ve \alpha},t,h,\eta}{k}{}$ is MDS.
\end{theorem}

\begin{proof}
Let $\eta \in \mathbb{F}_q \setminus \mathbb{F}_s$.
Let $\mathcal{I} \subseteq \{1,\dots,n\}$ be an index set with $|\mathcal{I}|=k$ and $\ve A_\mathcal{I}$ be the corresponding matrix as in \cref{lem:MDS_property_one_twist}.
Using elementary row operations, we can bring $\ve A_\mathcal{I}$ into lower triangular form
with diagonal elements $\eta^{-1} + T,1, \ldots, 1$ for a certain $T \in \mathbb{F}_q$. Using that $\sigma_i \in \mathbb{F}_s$ for all $i$ (since the same holds for all $\alpha_i$), we conclude that in fact $T \in \mathbb{F}_s$.
Since $\eta \not\in \mathbb{F}_s$, the diagonal elements of the triangular form of $\ve A_\mathcal{I}$ are all non-zero, implying that $\ve A_\mathcal{I}$ is regular.
\end{proof}

\section{Non-GRS MDS Twisted Codes}

\noindent
Since GRS codes are always MDS and well studied, we will in this section show that most of the twisted codes are not equivalent to an RS code. The main result is the following theorem.

\begin{theorem}\label{thm:not_GRS_one_twist}
Let $\alpha_1,\dots,\alpha_n \in \mathbb{F}_q$ and $2 < k < n-2$.
Furthermore, let $\mathcal{H} \subseteq \mathbb{F}_q$ satisfy that the twisted code $\TRS{{\ve \alpha},t,h,\eta}{k}{}$ is MDS for all $\eta \in \mathcal{H}$.
Then there are at most $6$ choices of $\eta \in \mathcal{H}$ such that $\TRS{{\ve \alpha},t,h,\eta}{k}{}$ is equivalent to an RS code.
\end{theorem}
\begin{proof}
Since $\TRS{{\ve \alpha},t,h,\eta}{k}{}$ is an MDS code, it has a generator matrix of the form $\ve G = [\ve I \mid \ve A]$. Equivalently, there exist polynomials $\fpolyi{i} \in \mathcal{V}_{k,t,h,\eta}$ such that for all $i$ and $j$ in $1,\ldots, k$ it holds that $\fpolyi{i}(\alpha_j) = 1$ if $i=j$ and $\fpolyi{i}(\alpha_j) = 0$ if $i \neq j.$ Further for $j>k$ we have $\fpolyi{i}(\alpha_j) = \ve A_{i,j-k}$. In particular, since $\fpolyi{i} \in \mathcal{V}_{k,t,h,\eta}$, the $(i,j)$th entry of $\ve A$ is of the form $\ve A_{i,j}=c_{i,j}+d_{i,j}\eta$ for certain $c_{i,j},d_{i,j} \in \mathbb{F}_q.$

Now we use item \ref{itm:GRS_characterization_3} of Theorem~\ref{thm:GRS_characterization} to derive an upper bound on the number of choices of $\eta$ such that $\TRS{{\ve \alpha},t,h,\eta}{k}{}$ is equivalent to an RS code. Let us consider the minor $M$ of the first three rows and columns of $\ve{\tilde{A}}$. Then $\TRS{{\ve \alpha},t,h,\eta}{k}{}$ is not equivalent to an RS code if $M$ does not vanish. However, since $\ve{\tilde{A}}_{i,j}=\frac{1}{c_{i,j}+d_{i,j}\eta}$, this minor is of the form
\begin{align*}
M=\frac{p(\eta)}{\prod_{i,j=1}^3(c_{i,j}+d_{i,j}\eta)},
\end{align*}
where $p(\eta)$ is a polynomial in $\eta$ of degree at most $6$. Hence $M$ can vanish for at most six values of $\eta$, which implies the theorem.
\end{proof}

\cref{thm:not_GRS_one_twist} directly implies the existence of non-GRS MDS twisted codes for many field sizes, as appears from the following corollaries.

\begin{corollary}
  \label{cor:existence_non_GRS_star_plus_codes}
   Suppose $(\mathbb{F}_q^*, \cdot)$ has a non-trivial subgroup $G$ such that $|\mathbb{F}_q^* \setminus G| >6$. Then for any $n, k$ with $2 < k < n-2$ and $n \leq |G|$ there exists a non-GRS MDS $(*)$-twisted  code. Similarly if $(\mathbb{F}_q,+)$ has a non-trivial subgroup $V$ such that $|\mathbb{F}_q \setminus V|>6$, then for any $n, k$ with $2 < k < n-2$ and $n \leq |V|$ there exists a non-GRS MDS $(+)$-twisted  code.
\end{corollary}

\begin{corollary}\label{cor:existence_non_GRS_CTRS_one_twist}
Let $\mathbb{F}_s \subsetneq \mathbb{F}_q$ with $|\mathbb{F}_q \setminus \mathbb{F}_s|>6$. Let $2 < k < n-2$ and $n\leq s$.
Then, there exists $\eta \in \mathbb{F}_q \setminus \mathbb{F}_s$ such that $\TRS{{\ve \alpha},t,h,\eta}{k}{}$ is MDS but not equivalent to a GRS code.
\end{corollary}

\begin{remark}\label{rem:existence_non-GRS_single-twisted}
From $q \geq 13$, non-GRS $(*)$-twisted  or $(+)$-twisted  codes of length $\lceil(q+1)/2 \rceil$ is guaranteed by \cref{cor:existence_non_GRS_star_plus_codes}.
If $q=p^m$ is a prime power with $p>3$ and $m>1$, then the restriction $|\mathbb{F}_q \setminus \mathbb{F}_s|>6$ of \cref{cor:existence_non_GRS_CTRS_one_twist} is fulfilled for all $\mathbb{F}_s \subsetneq \mathbb{F}_q$.
\end{remark}

\section{Computer Searches}
\label{sec:Computer_Searches}

In this section, we present exhaustive computer searches for twisted codes over small field sizes.
Since we are most interested in non-GRS codes, we only perform searches for $\eta \neq 0$ and $\min\{k,n-k\}>2$ (cf.~\cref{cor:min_k_n-k_>_2}).
The compututations were carried out using SageMath v7.4 \cite{stein_sagemath_????}.
The full results and the source code can be downloaded from \url{http://jsrn.dk/code-for-articles}.

\subsection{Number of $(*)$-Twisted  Codes}

We counted all $(*)$-twisted  codes over $\mathbb{F}_q$ for $q \leq 19$ and all $(+)$-twisted  codes over $\mathbb{F}_q$ for $q = 16$ and $q = 49$, i.e., the number of sets $S$ and $\eta$'s that fulfill the conditions of \cref{def:startwisted} respectively \cref{def:plustwisted}.
Moreover, we have determined how many of the resulting codes are inequivalent and which of those are not GRS codes.

As predicted, for odd $q$, there are $(*)$-twisted  codes of length up to $\frac{q+1}{2}$ and arbitrary $k$. It also turns out that almost all $(*)$-twisted  codes are non-GRS.
In particular, there is exactly 1 $(*)$-twisted  $[11,6,3]$ code which is not GRS, even though \cref{cor:min_k_n-k_>_2} did not guarantee this.

\cref{tab:star_twisted_computer_search_example} exemplifies the results for $q=19$.

\begin{table}[h]
\caption{Number of $(*)$-twisted  codes over $\mathbb{F}_{19}$ (Total / Inequivalent / Non-GRS).}
\label{tab:star_twisted_computer_search_example}
\centering
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{c|c|c|c|c|c}
$n\backslash k$ & $3$ & $4$ & $5$ & $6$ & $7$ \\
\hline
$6$ & $1974/73/67$ &   &  &   &     \\
$7$ & $1092/67/67$ & $1092/63/63$  &  &   &     \\
$8$ & $405/25/25$ & $405/25/25$  & $405/25/25$ &   &     \\
$9$ & $90/7/7$ & $90/6/6$  & $90/6/6$ & $90/7/7$  &     \\
$10$ & $9/2/2$ & $9/1/1$  & $9/1/1$ & $9/2/2$  & $9/1/1$
\end{tabular}
\end{table}

\subsection{Comparison with Roth--Lempel Codes}

Roth and Lempel \cite{roth_construction_1989} gave a construction of non-GRS MDS codes: given $S \subset \mathbb{F}_q \cup \{\infty\}$ with $n = |S|$ which is \emph{not} a $(k-1)$-sum generator of $(\mathbb{F}_q, +)$, it produces an $[n,k]$ MDS code\footnote{%
  We say that a set $S$ containing $\infty$ is a $k$-sum generator if $S \setminus \{\infty\}$ is a $k$-sum generator.
  Note that for comparison with our codes, we relax the construction of \cite{roth_construction_1989} by allowing $S$ which does not contain $\infty$.
  }.
Roth and Lempel point out, similar to our \cref{def:plustwisted}, that e.g.~subgroups of $\mathbb{F}_q$ will give such non-$(k-1)$-generators.
When $q$ is even, these explicit constructions allow codes in a similar range as $(+)$-twisted  codes.
When $q$ is odd, their construction is much worse; for $q$ an odd prime, they remark in \cite{roth_t-sum_1992} that asymptotically their construction has at most length $\frac q k(1 + o(1))$.

For small $q$ one can exhaustively search for all non-$(k-1)$-sum generators, however, and thereby produce all Roth--Lempel (RL) codes.
We have done this for some parameters with the aim of determining how often $(*)$-twisted  or $(+)$-twisted  codes are equivalent to RL codes; especially for the $(+)$-twisted  codes where the possible range of parameters largely coincides.

Our computer searches indicate that the code families are largely independent.
We give three examples:

For $(q, n, k) = (13, 7, 3)$, there are 35 inequivalent RL codes, while there are 2 $(*)$-twisted  codes; 1 code is in both sets.
There are no $(+)$-twisted  codes of these parameters, but there are 8 twisted codes with $(t,h) = (1,k-1)$; 2 of these are RL codes.

For $(q, n, k) = (16, 8, 5)$, there are 186 inequivalent RL codes, while there are 9 inequivalent $(+)$-twisted  codes.
These codes are all different.
There are 83 twisted codes in total with $(t,h) = (1,k-1)$, and 10 of these are RL codes.

For $(q, n, k) = (23, 12, 5)$, there are no RL codes, while there is 1 equivalence class of $(*)$-twisted  codes.

\subsection{Length $\approx q/2$ Codes with ``Exotic Twists''}

$(*)$-twisted  and $(+)$-twisted  codes are explicit subclasses of the cases $(t,h) = (1,0)$ respectively $(t,h) = (1,k-1)$ which allow codes of length $\approx q/2$ for odd respectively even $q$.
A natural question is if similar long MDS codes are possible for twisted codes of other $(t,h)$.

We have no explicit construction, but exhaustive search indicates a resounding 'yes': in fact, for any $q \leq 19$ we verified that for almost \emph{any} choice of $(t, h)$ there is an $[n,k]$ twisted MDS code for $n = \lfloor q/2 \rfloor$ and $3 \leq k \leq n-3$, with the only exceptions being $(t, h) = (1, k-1)$ which fails for $(q,k) = (17, 4)$ and $q = 19$ and any $k$.

The non-existence for $(t, h) = (1, k-1)$ should indeed be expected for high enough $k$ due to \cref{lem:k_sum_gen_plus_MDS} and the asymptotic upper bounds on $M(k, \mathbb{F}_q)$ discussed in \cite{roth_t-sum_1992}.
However, it is surprising and intriguing that for all other $(t,h)$, there seem to be long twisted MDS codes.

\subsection{Counting Twisted MDS Codes}

\cref{tab:Search_Equivalence_Classes} enumerates all MDS twisted codes for $q \leq 13$, the number of equivalence classes as well as non-GRS equivalence classes.
We see that many parameters often lead to equivalent codes: e.g. for $(q,n,k)=(13,7,4)$, each equivalence class is obtainable by $\approx\!\!1200$ parameters on average.
This might be due to algebraic symmetries in the parameter choices, which is an interesting question to investigate.
However, most MDS twisted codes are non-GRS.
Note that in most of the cases, we can construct codes of length $q-1$ for $k \in \{3,n-3\}$.
Apart from Glynn's code and its dual \cite{glynn1986non}, we find only a single new code of length $q$: $(q,n,k)=(8,8,5)$ and constructible as e.g.~$\TRS{{\ve \alpha},1,4,1}{5}{}$ with ${\ve \alpha} = \mathbb{F}_8 \cup \{ \infty \} \setminus \{ 1 \}$.
This code is also equivalent to a Roth--Lempel code.

{
\newcommand{\Count}[3]{#1$/$#2$/$#3}
\newcommand{}{}

\begin{table}[b]
\caption{Counting MDS Twisted Codes (Total/Inequivalent/Non-GRS. Blank = 0/0/0)}
\label{tab:Search_Equivalence_Classes}
\centering
\renewcommand{\arraystretch}{1.2}
\setlength{\tabcolsep}{1pt}

\tiny
\begin{tabular}{@{}c|c||c|c|c|c|c|c|c|c@{}}
$q$ & $n$ & $k=3$                   & $4$                   & $5$                     & $6$                  & $7$                & $8$               & $9$ \\
\hline \hline
7
    & 6   & \Count{38}{3}{2}        &                       &                         &                      &                    &                   &     \\
\hline
8
    & 6   & \Count{406}{5}{4}       &                       &                         &                      &                    &                   &     \\
    & 7   &               & \Count{63}{2}{1}      &                         &                      &                    &                   &     \\
    & 8   &               &             & \Count{14}{2}{1}        &                      &                    &                   &     \\
\hline
9
    & 6   & \Count{2374}{7}{5}      &                       &                         &                      &                    &                   &     \\
    & 7   & \Count{216}{3}{3}       & \Count{332}{3}{2}     &                         &                      &                    &                   &     \\
    & 8   & \Count{4}{1}{1}         & \Count{40}{1}{1}      & \Count{36}{1}{1}        &                      &                    &                   &     \\
    & 9   &               & \Count{4}{1}{1}       & \Count{4}{1}{1}         &            &                    &                   &     \\
\hline
11
    & 6   & \Count{32518}{15}{11}   &                       &                         &                      &                    &                   &     \\
    & 7   & \Count{6286}{21}{19}    & \Count{8554}{20}{18}  &                         &                      &                    &                   &     \\
    & 8   & \Count{585}{15}{15}     & \Count{160}{7}{6}     & \Count{960}{9}{7}       &                      &                    &                   &     \\
    & 9   & \Count{40}{3}{3}        &             & \Count{20}{1}{0}        & \Count{135}{1}{0}    &                    &                   &     \\
    & 10  & \Count{2}{1}{1}         &             &               &            & \Count{22}{2}{1}   &                   &     \\
\hline
13
    & 6   & \Count{216722}{26}{21}  &                       &                         &                      &                    &                   &     \\
    & 7   & \Count{71618}{80}{75}   & \Count{98430}{80}{75} &                         &                      &                    &                   &     \\
    & 8   & \Count{11164}{165}{160} & \Count{5176}{98}{93}  & \Count{26916}{139}{134} &                      &                    &                   &     \\
    & 9   & \Count{1110}{32}{31}    & \Count{41}{4}{3}      & \Count{381}{8}{5}       & \Count{5424}{24}{21} &                    &                   &     \\
    & 10  & \Count{138}{4}{4}       &             &               & \Count{93}{3}{0}     & \Count{1167}{4}{1} &                   &     \\
    & 11  & \Count{24}{1}{1}        &             &               &            &          & \Count{254}{1}{0} &     \\
    & 12  & \Count{2}{1}{1}         &             &               &            &          &         & \Count{26}{2}{1}
\end{tabular}
\end{table}
}

\section{Decoding of Twisted Codes}

A simple decoding paradigm is possible for twisted codes:
guess the value of the hook coefficient $a_h$, and then apply an $[n, k]$-RS decoding algorithm on $r - \ev{\eta a_h x^{k-1+t}}{{\ve \alpha}}$, where $r$ is the received word.
If the twisted code is over $\mathbb{F}_q$, this will apply the RS decoder $q$ times.
This works with errors, erasures, soft-decision, list-decoding, etc.
In particular, we have (cf.~\cite{justesen_complexity_1976}):

\begin{theorem}
  An $[n,k]$ twisted code over $\mathbb{F}_q$ can be decoded up to half the minimum distance in complexity $\Osoft(qn)$.
\end{theorem}

Note that even if the twisted code is not MDS, this approach still gives a list-decoder up to $(n-k+1)/2$: collect the codewords obtained from each guess of $a_h$, and the correct codeword is on the resulting list if the number of errors is at most the decoding radius of the RS decoder.

\section{Conclusion}

We have introduced twisted Reed--Solomon codes, a new class of $\mathbb{F}_q$-linear codes, and demonstrated that for $q \ge 7$ the class contains many new non-GRS MDS codes.
We singled out two explicit subclasses, $(*)$-twisted  and $(+)$-twisted  codes, with which we can construct MDS codes of length at least $q/2$ for any field size $q$, and that for most field sizes, most of these codes will be non-GRS.
Using computer searches we demonstrated that there seems to be many other and longer MDS, non-GRS twisted RS codes.

\bibliographystyle{IEEEtran}

redpajama_set_name: RedPajamaArXiv


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I have picky eaters in this house. They get it from their daddy. I have decided that each week, we will try to learn to like something new. This week, we are learning to like celery.
How did the boys respond to this today?
That was much easier than I anticipated. Now to think of what we'll work on for next week. Perhaps broccoflower might be pushing it... at least for now (and especially since I don't even know if *I* will like it!).
Did your parents make you learn to like certain foods when you were growing up? While my dad didn't exactly make us learn to like cow tongue when we were little, he did force us to try it! Blech!!! Although I do believe that my sister Lele threw a fit or maybe ran away or something to avoid having to eat it. She was always smart like that. There are definite drawbacks when grandma and grandpa have a farm and cows.


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text:
Catalyzing Commercialization: A Graphene-Based Sensor Monitors Water Quality in Real Time
The devastating water crisis in Flint, MI, brought into focus the widespread issues plaguing U.S. water supply systems. As the water infrastructure continues to age, distribution pipes are deteriorating, potentially leaching toxic heavy metals and other contaminants into drinking water. Upgrading U.S. water systems will take years and trillions of dollars. In the meantime, to safeguard public health, it is crucial to monitor heavy metals and other contaminants throughout the water supply — particularly at the point of use.
Existing in situ water testing technologies are inadequate. Laboratory-based technologies can quantitatively measure aqueous heavy metal ion content down to parts per billion levels. However, these methods are costly (about $25–$50 per test) and require days for sample analysis to be completed, making them unsuitable for regular onsite detection. Qualitative detection methods are suitable for the field, relatively inexpensive (e.g., colorimetric test kits cost about $10/kit) and quick, but they are unreliable and provide inadequate sensitivity.
With funding from the National Science Foundation (NSF), researchers at the Univ. of Wisconsin-Milwaukee (UWM) and NanoAffix Science, LLC, are developing a sensitive, selective, and cost-effective micro-sized sensor for the quantitative, real-time, in situ detection of heavy metal ions in water. The sensor is a field-effect transistor with reduced graphene oxide (rGO) as the sensing channel and specific molecules attached to gold nanoparticles as probes. The sensor shows promise for detecting a wide range of analytes, including heavy metal ions, bacteria, and nutrients in water.
▲ A handheld device that includes a sensor enclosed in a test cell and attached to a digital meter can measure lead well below 15 ppb. The sensing chip is a field-effect transistor, with reduced graphene oxide (rGO) as the sensing channel and specific molecules attached to gold nanoparticles as probes. When heavy metals contact the probes, they react and change the conductivity of the rGO, which can be converted into the concentration of the heavy metal of interest.
Here's how the sensor works. The conductivity of the rGO channel changes when chemicals/analytes bind to the probes anchored on the gold nanoparticles, e.g., through chelation reactions with heavy metals. The change in conductivity (measured in resistance) is proportional to the quantity of metal ion the sensor is measuring. The probe is designed to target a specific analyte.
The new technology allows real-time detection (no sample preparation is required) of contaminants with unprecedented sensitivity and selectivity in field settings. A handheld device is suitable for single-point testing, or the technology can be integrated into existing water equipment to provide continuous monitoring.
The researchers are collaborating with A.O. Smith, Badger Meter, and Baker Manufacturing to commercialize the sensing technology. For instance, A.O. Smith, a manufacturer of water heaters and water-treatment products, is interested in integrating the sensor into its water purifiers.
"This sensing technology gives us the ability to make sure contaminants are removed by our filtration products," says Robert Heideman, senior vice president and CTO at A.O. Smith.
A prototype of the handheld device, consisting of the sensor housed in a test cell and attached to a digital meter, showed promise for rapid and selective detection of lead ions in drinking water. In the lab and in point-of-use field tests, it was able to measure lead well below the 15-ppb level required by the U.S. Environmental Protection Agency (EPA). In addition, the sensor detected Pb2+ within seconds — a feature promising for real-time monitoring.
The developers say that the availability of the handheld device, with an estimated retail cost of $60/unit (approximately $20 for the one-time sensor chip and $40 for the reusable meter), will contribute to improving the safety of drinking water and thus enhance public health.
The UMW and NanoAffix team is now testing the device in practical environments for real-time, in situ operation. Future work will focus on improving the performance of the device for continuous water monitoring and the detection of bacteria and nutrient contaminants, as well as on scaling up the process to manufacture these devices to reduce their cost.
This technology was funded through the NSF Partnerships for Innovation Program.
This article was prepared by the National Science Founda tion in partnership with CEP.
A Bright Spot for Energy Production
Recently, I had the opportunity to view a great swath of the U.S. in less than six hours. Flying west from New York to California, I spotted major cities and snow-capped mountain ranges, but what stood out to me the most was an enormous circular arrangement of mirrors all facing one central point. From my aerial vantage, I recognized it as a concentrating solar power (CSP) plant, and later learned it was likely the Crescent Dunes CSP plant in Nevada.
Concentrating Solar Thermal Overview
Concentrating solar thermal (CST) technology can be used to generate electricity, desalinate water, and more. Find out how CST works and the role chemical engineers can play.
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Thermochemical energy storage could be the key to widespread concentrating solar power (CSP) deployment.
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Adsorption can be used to treat waste streams or purify valuable components of a feed. This article describes both equilibrium and mass-transfer considerations, and reviews the fundamentals of adsorption system design.
Making Sense of Laboratory Fire Codes
Laboratory owners are faced with a difficult task when building a new lab or renovating an existing one — two fire-prevention codes specify different restrictions on the quantities of hazardous materials that can be used and stored in that facility.
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Stack testing, material balances, emission factors, and engineering equations are the most common methods that can be used to estimate air emissions.


Unlike ecological environments, fast-growing urban habitats do not just turn green naturally – they need to be made eco-friendly through meticulous care, and few cities illustrate that truism better than Bangkok.
Like other metropolises in a rapidly urbanizing region, Thailand's capital has over the past decades been experiencing unprecedented growth both in the ever-increasing number of its residents and its sheer geographical size. Yet much of the city's development has been helter-skelter, without proper planning and foresight. Nor has it come without severe environmental costs and social disadvantages.


text:
La Realtà degli Altri
This work was developed in response to the archive of the house-museum Casa Testori, dedicated to the art historian, collector, writer, dramaturg and painter Giovanni Testori.
"When I first visited Casa Testori I noticed a series of books about African art. I was told that Giovanni Testori's interest in this topic emerged towards the end of his life. There seemed to be very little information on what might have moved Testori to be fascinated by this, except for three articles published on Il Corriere della Sera. Within three exhibition reviews that he penned in 1989-90, Testori offers some insights into his thoughts, and they are, I believe, quite refreshing. While major Italian museums and galleries were swept by a newfound fascination with African artefacts at the time, Testori questioned the assumptions and epistemological violence inflicted by the Western museum's display and knowledge system onto these oeuvres belonging to a different (Other) culture. I was aware that this late interest pales in comparison with the magnitude of Testori's life's work and so in the installation I have tried to put these lesser-known writing in relation to one of Testori's most characterising and enduring activities, that of art historian. I was captured by two boxes full of black and white photographs of paintings (mostly dating between the 16th and 18th century I believe), that had not been catalogued yet. They were used by Testori for the purpose of authenticating them. Upon finding, amongst them, a picture of Testori's teacher, Roberto Longhi, with a magnifying glass in his hand, reminiscent of a modern-day detective, my mind went immediately to Carlo Ginzburg's seminal essay Morelli, Freud and Sherlock Holmes: Clues and Scientific Method. By linking the work of these three figures (respectively a connoisseur art historian, the father of psychoanalysis and the quintessential detective, albeit fictional) Ginzburg sketched a genealogy of what he calls the "conjectural paradigm" – namely, an "evidential approach" which "builds knowledge of the whole from its parts" through the art of observation and deduction, rooted in symptomatology (medical semiotics). As I looked through the boxes full of black and white images, with Testori's words on African art still resonating in my mind, I started to look out for clues as to how (and whether) non-Western subject were featured and represented. I could only find four images featuring non-white subjects. Inspired by Morelli's illustrations of ears, hands and other body parts lifted from paintings in order to determine a signature execution style capable of assigning authorship, I have isolated these subjects to reveal their placement within the compositions and how they were portrayed. These materials were, finally, put in relation with a third element: a letter addressed to Testori requesting the authentication of a painting, whose owner claimed might have belonged to the Lombard painter Daniele Crespi. There is a striking story surrounding this very image, that claims that Crespi killed a man and used his corpse as a prop, in order to be able to portray the most realistic account of the body of Christ after crucifixion. I thought this story could enable a reflection on the way the Western aesthetic paradigm of realism – and its hegemonic status worldwide, especially over cultures deemed as 'primitive' – brings with it an aura of violence, whether physical (in such extreme, anecdotic cases) or epistemological."
From the exhibition catalogue "Se la realtà non è solo un fotogramma", curated by Alessandro Castiglioni.
info ≈
contact ≈
news ≈
GADDAFI IN ROME
≈ Gaddafi in Rome: Notes for a Film
≈ Gaddafi in Rome: the Expanded Script
≈ Sight Unseen
A BOMB TO BE RELOADED
≈ Chapter 0 (Resistant Archives)
≈ Chapter 1 (Resistant Musings)
≈ Chapter 2 (Resistant Voices)
RADIO GHETTO RELAY
Radio Ghetto Relay ≈
OutCross ≈
≈ Outcross (Black Mediterranean)
NEGOTIATING AMNESIA
Negotiating Amnesia ≈
≈ Notes on Historical Amnesia pt.2
Notes on Historical Amnesia ≈
≈ La Realtà Degli Altri
≈ Archive-as-Method
≈ Shot/Countershot
≈ Mnemoscape
ALESSANDRA FERRINI


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Falling Back Into TITANFALL With Latest Patch
I've more or less fallen out of love with Respawn's Mechwarrior meets Call of Duty hybrid shooter, Titanfall, over the few months since its release. Most of my reasons are much the same asCharles' experience. Simply put, Titanfall failed to hold my attention. This isn't a total reflection on the end product as Titanfall is a pretty fun game overall, yet the kibbles and bits that keep me interested could not whet my appetite.
Part of this is due to my gaming habits whereby I'll play a game or two to death and then quickly lose interest. The other part is that, despite Titanfall coming out swinging with a couple of neat twists on the pull right trigger and then pull left trigger genre there simply isn't enough there.
In a recent patch, Respawn has tried to mix things up a bit to keep players coming back. Two new modes have been revealed: "Marked for Death" and "Wingman Last Titan Standing." Marked is available now while Wingman LTS will debut a little later. However, there's a big asterisk next to these modes meaning that they're available for a limited time only. So act fast.
At any rate, I decided to re-enlist to see how the new mode plays.
Marked for Death is a welcome change to the traditional deathmatch style. Basically, a randomly-chosen pilot from each team is tagged "it" and needs to be killed before your team's marked pilot is iced. This creates an intense atmosphere of paranoia when you're tagged. It also creates wonderful hotspots for a bit of tactical thinking for each team: do you hang back and guard your teammate or do you charge forward to get the jump on theirs? The modes has only been out for a couple of weeks, so strategies tend to vary. Really, though, it's just more localized deathmatch variant.
Image taken from the Xbox One version of Titanfall.
As with any twitch reflex game, map knowledge is a must and knowing how to deftly maneuver into a safe place can make or break your time as being marked. It's also a markedly more aggressive mode. Marked targets are highlighted until they die, mocking your HUD in defiance. This gives you a sense of purpose that is missed from standard Attrition in which everyone kinda just does their own thing. The overall flow keeps the same speedy pace as the other modes, but the added direction makes matches incredibly fast.
However, I'm still unsure if a steady rotation of modes will keep players in the game. Titanfall doesn't exactly differentiate itself from the CoD crowd that much. Furthermore, the new Wingman mode doesn't seem that much of a departure from the original as it takes the same idea – destroy the other players before they kill you – except reduces the players to teams of two. I guess we'll see how it plays in the next update.
I'm hoping that Respawn can bring in some more unique modes. I'm sure at some point there will be a CTF variant and maybe even a Sabotage mode as well. But I'd like to see more ideas to push Titanfall away from the lineup of similar titles. Specifically, I'd love for them to try out a Unreal's Onslaught mode or maybe even attempt to create an FPS/MOBA monstrosity. The already have the AI units, they could add in more capture points or something and make something extremely tactical and fun. They could even ape the upcoming Evolve by having four pilots take on a super buffed Titan. The possibilities are virtually limitless. If Respawn can grant new modes without additional cost on the player side, then perhaps some of those glass half full feelings about the amount of in game content could hushed.
Marked for Death and some of the other updates – like the rebalancing of weapons and some more unlocks and new cards – are giving me some trigger happy fingers. If Respawn can keep up some momentum, perhaps Titanfall will appear in my regular rotation again. Certainly, they've heard from the community and are thankfully responding in kind. Now let's just see what they're able to do with the feedback.
Respawn EntertainmentTitanfallXbox One
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59 lakh fake voters in Andhra: YSR Congress
February 21, 2019 by neha
Amravati: With an Election Commission (EC) team visiting Andhra Pradesh to audit the state voter list, the YSR Congress Party (YSRCP) on Thursday instensified its attack on the ruling Telugu Desam Party (TDP), claiming over 59 lakh fake voters had been added to the list at its behest.
Out of the total 3.69 crore names on state voter rolls, over 59 lakh (16 per cent) are invalid or duplicate, the YSRCP said at a press conference on Thursday and alleged that the TDP government was doing everything in its power to subvert democracy and steal the elections.
The party also released a detailed survey pointing out 59,18,631 voters under 12 different categories such as duplicate, illegal, invalid, bogus and demographically similar entries. It said the number of such voters was 52.7 lakh in September 2018.
The party further said that over 20 lakh people have votes in both Andhra Pradesh and Telangana.
YSRCP chief Y.S. Jagan Mohan Reddy had on February 4 met the Chief Election Commissioner to bring to his notice discrepancies in the state voter list and requested him to intervene in the matter to ensure free and fair elections.
YSRCP Legal Cell President P. Sudhakar Reddy then filed a Public Interest Litigation (PIL) in the High Court seeking directions to the EC to rectify the discrepancies.
The High Court has sought a clarification from the Election Commission in the matter after an audit of the list.
Following the directive, a 10-member EC team, led by former Deputy Election Commissioner Vinod Zutshi, arrived in the state on Wednesday to conduct an audit of electoral rolls.
Categories Andhra Pradesh, News Tags andhra, Jagan mohan reddy, YSR Congress
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The 15 Best Destinations to Visit in Peru
Huaraz | © Madeleine Deaton / Flickr
Brandon Dupre
You may know all there is to know about Cusco and the legendary Machu Picchu, but Peru is home to so many other spectacular sights that can easily go unrecognized. From the jungle to the Andes mountains, Peru is one of the most diverse countries in the world, and has a rich cultural history and tradition. Here are 15 must-see destinations that'll take you on an unbeatable adventure through this unique country.
Lima is a must-visit, especially considering the food renaissance that is taking place in the city these days. The "City of Kings" now has two restaurants that rank inside the top 10 in the world, and even the little hole-in-the-wall restaurants stay true to quality Peruvian gastronomy. With beautiful places such as the Plaza de Armas and the bohemian neighborhood of Barranco to visit, Lima has a lot to offer.
Lima Cathedral | © Art DiNo / Flickr
Once you come, you'll never want to leave – that's what everyone who has ever been to Huaraz will tell you. With the Cordillera mountain range in its backyard, this mountain town has gained a reputation as the place to go backpacking in Peru. There are seemingly endless options that are sure to amaze.
Markawasi
The inscrutable stone figures are a must-visit and are only six hours from Lima. Hike high into the Andes mountains and discover the mystery that is the stone faces and figures of Markawasi, but be careful – going from sea level in Lima to the Andes too quickly will give anyone altitude sickness. Remember to take it slow.
This isn't just another beach town along the coast of Peru. Huanchaco is steeped in history and tradition, which is evident as you take a walk done the boardwalk and see the fishermen in their reed boats dating back to time immemorial.
Caballitos de totora, Huanchaco | © fromhomeparacasa / Flickr
The beach capital of Peru, Mancora has everything you want in a beach town: nightlife, surf, sunny skies, and warm weather. With some of the most delicious and fresh food in all of Peru, this town seems to have everything.
Mancora | © Jonathan Hood / Flickr
The town itself is small and quiet, but people come here for the hiking. Just two hours outside this mountain town are two of the most spectacular sites in Peru: Gocta Waterfall, hidden in the mountains; and Kuélap, a walled city high in the mountains that was built by the Chachapoyas culture.
Kuélap ruins | © Hanumann / Flickr
The city of Iquitos, located smack dab in the Peruvian jungle, is one of the more unique cities in Peru. Unreached by any roads, this town has an Amazon vibe that is not found anywhere else in the country, and it makes the perfect launching-off point for exploring the Amazon and for the ayahuasca retreats that take place close by.
The capital of the Incas is now the capital of tourism in Peru, and for good reason. Cusco is a remarkable city, surrounded by ruins and built on Inca foundations, with fascinating colonial architecture.
The hippy town of Pisac in the Sacred Valley offers beautiful ruins on top of a mountain and one of the best markets in Peru. You'll also find all of your shamanic tools here, such as rapé (sacred shamanic snuff medicine) and rapé pieces.
Pisac | © AHLN / Flickr
The absolute must-see destination in Peru is, of course, Machu Picchu. It is one of those places that you won't believe until you have seen it. The Inca citadel in the sky is an architectural marvel that is likely to leave you speechless.
The small town of Ollantaytambo in the Sacred Valley is a perfect example of Inca street planning and design. With narrow cobblestone streets and ruins that cling to the cliffside above, Ollantaytambo will make you feel that you are stepping back in time.
Ollantaytambo | © jeremynelms / Pixabay
The highest navigable lake in the world is also home to the Uros people, a culture that continues to live on islands made out of reeds.
The whitewashed city of Arequipa is home to one of Peru's most beautiful Plaza de Armas and some of the country's best food, which is saying a lot. Colca Canyon, which makes the Grand Canyon look small, is only a short ride away.
The unexplained lines that are etched in the landscape are worth a trip to the Nazca. You can take a short plane ride above the lines so you can make out the shapes and come up with your own theory as to how they were created.
Nasca lines | © monikawl999 / Pixabay
This desert oasis is only a short ride from Ica. It's surrounded by rolling rills of sand, and you really do feel as if you are stranded while staying at this tiny pond lake in the Peruvian desert. Take a day trip to explore the sand dunes on ATVs; you can even surf them.


It is a real war! And here you will have to be a tank crew member to defeat your enemies! Use the arrow keys to move your tank and use the mouse to aim at the enemy tanks. Click the left mouse button to shoot. Good luck!


Unix v7, released for the PDP-11 line of minicomputers in 1979 is one of computings classics. Just a year before The C Programming Language had been released in its first edition by Bell Labs. In this release is the first appearance of pcc, the Portable C Compiler which began C's ascendancy as the de facto systems language in use today. This is the same year that Bill Joy stepped down as lead developer of vi. Many of the programs you use on a daily basis can trace their ancestry here.
The emulator we'll be using, SIMH, was written in 1993 to preserve the dying minicomputer culture that spawned Unix, VMS and other operating systems of the seventies and eighties. As a result, most of the images to boot are old, and their instructions have not kept up with the times. This is a short tutorial to correct this for V7.
SIMH can be downloaded from its website for free. Precompiled executable are available for Windows, if you use a Unix or Unix-like platform such as Macintosh or Linux downloading from the site means compiling the sources yourself. However it is highly likely that alternative, more convenient means exist.
After you've installed the emulator it's time to boot a copy of Unix.
As a further disclaimer the author has only tried this on Linux, and cannot speak in further detail for Windows or Macintosh operating systems.
This will help keep boot files and other artifacts in a contained clean workspace. Unpack the file.
Set the teletype output to seven bits.
Before you can start using your new Unix install you must make a /tmp directory, if you do not do this essential programs such as Portable C Compiler (pcc) will fail to execute. Furthermore the default user included with the system, the deceased Dennis MacAlistair Ritchie is missing a home directory. Kudos goes to Markus Wandel for writing a tutorial about this in 2002 describing these steps.
First we want to create a tmp directory.
Then press control-D to exit the current prompt and login as dmr, the account requires no password.


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Council sharing car park revenue with Lincoln cinema developers
Photo: File/The Lincolnite
The City of Lincoln Council is sharing revenues from the Lucy Tower car park with the developers of the Odeon cinema on Brayford Pool.
City Council Leader Ric Metcalfe confirmed the information to The Lincolnite during a weekly Q&A session on the council's Facebook page.
He confirmed 20% of parking revenues between 6pm to midnight and from midday on Sundays go to the developers of the Odeon Lincoln cinema.
This is part of an agreement between the council and the developers of the building as a long-term incentive to bring the Odeon to the city centre.
Councillor Ric Metcalfe said: "We did enter into an agreement as an incentive to secure the Odeon's presence in the city centre.
"This has been a big boost to Lincoln's night-time economy and the agreement only relates to parking between 6pm to midnight and from noon on Sundays."
— Update on February 6, 2014: The City of Lincoln Council got back in touch to say that the revenue is shared with the developers of the building the cinema is in, and not the cinema chain itself. This story has been edited to reflect that.
The City of Lincoln Council said in a further statement: "The income received, less the costs of providing the service, at Lucy Tower Street car park between 6pm and midnight Monday to Friday and from noon to midnight on Sundays goes to the development company and not the Odeon or any other Brayford business.
"This agreement was reached in 2000 as part of the council's efforts to regenerate the Brayford waterfront. A cinema was seen as integral to enticing other business to open in this area."
Cinema move to city centre
The Odeon Lincoln cinema on Brayford Pool
In November 1995, a six-screen Odeon cinema opened on Tritton Road, currently the bowling alley premises.
But in October 2001, the Tritton Road venue closed and was replaced by a nine-screen Odeon on Brayford Wharf, which opened later that month.
Finally in 2003, the Tritton Road premises was converted into a Superbowl 10-pin bowling alley, which still operates now.
Car park back in action
The Lucy Tower car park was built in 1973 and has 458 spaces.
Lucy Tower brings more than £850,000 per year in revenue for the City of Lincoln Council.
Last year the multi-storey car park was closed from March to December 2013 for major structural repairs.
The works cost around £2 million in total, and the council lost around £20,000 per week in parking fees while the car park was closed.
The car park was partially opened early for the Lincoln Christmas Market since December, and to finalise the work overnight closures are in place until February 20.
Expensive cinema
The Lincoln Odeon cinema remains one of the most expensive cinemas in the region, with a regular adult ticket costing £8.70 for a 2D film, or £10.70 for a 3D screening.
The Odeon in Hull charges £7.50 for 2D screenings and £9.50 for 3D.
Odeon Sheffield charges £8.20 for 2D and £10.20 for 3D screenings, while Odeon Newark charges £7.95 for 2D and £9.95 for 3D.
The Vue cinema in Scunthorpe charges £6.40 off-peak and £7.55 at peak times, with an extra £2 for 3D screenings.
According to WhatPrice.co.uk, the average price of a cinema ticket in the Midlands is under £5.50.
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