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Stretching on Romney's Fees
By Robert Farley
An ad from the Obama campaign exaggerates the truth about fees and tax hikes imposed by Mitt Romney when he was governor of Massachusetts.
It lists numerous fees hiked by Romney, but we found that a number of them are misleading — seeming to be more far-reaching and broad than they really are.
The ad also repeats a misleading claim about Romney cutting taxes "on millionaires like himself," and botches the figure for the revenue gained by Romney imposing higher fees and closing corporate tax loopholes.
The ad is part of a one-two ad punch from the Obama campaign, attacking Romney's record as governor of Massachusetts. Called "Mosaic," it is the latest ad using the theme: "Romney Economics: Didn't Work Then. Won't Work Now."
The ad begins with a quote from Romney when he was running to be Massachusetts governor in 2002: "I'm going to reduce taxes."
The narrator then says that, "As governor, Mitt Romney did cut taxes on millionaires like himself, but he raised taxes and fees on everyone else — $1.5 billion. Over a thousand fee hikes. On health care. On school bus rides. On milk. On driver's licenses. On nursing homes. On lead poisoning prevention. On meat and poultry inspection. On fishermen and gun owners. On nurses. On electricians. On hospitals. On funeral homes. On mental health services. On hospice care. On elevator repair."
The narrator's voice trails off, implying that he could go on and on.
We'll deal with all three claims in order.
Tax Cuts for Millionaires?
The first claim is that "as governor, Mitt Romney did cut taxes — on millionaires like himself." We looked at this same claim when it was made in a previous Obama campaign ad, and we found it wasn't as clear-cut as the ad would have viewers believe. More accurately, Romney opposed a plan to impose a capital gains tax retroactively.
The tax has a bit of a tortured history. In 2002, before Romney became governor, the Massachusetts Legislature enacted a package of tax increases to deal with a deficit crisis. The package included a capital gains tax that was slated to go into effect in May 2002. But that tax was challenged in court, and the state's Supreme Judicial Court ultimately struck down the tax, ruling that it was unconstitutional for the tax to go into effect halfway through the year. That left the state Legislature with two possible remedies: have the tax kick in on Jan. 1, 2003 — and refund eight months' worth of capital gains tax revenue — or make it retroactive to Jan. 1., 2002, adding another four months' worth of tax revenue. The Democratic Legislature decided to make it retroactive to the start of 2002.
Romney argued it was unfair for people who made decisions, like selling their home, based on the tax law in place at the time to be forced to pay retroactively. He proposed that anyone who paid the tax in 2002 get a rebate.
The Boston Globe's editorial page, the nonpartisan Massachusetts Taxpayers Foundation and — ultimately — the Democratic leaders of the state Legislature, agreed. Although wealthy people are disproportionately affected by capital gains taxes, what Romney did was much different than a straight tax cut for millionaires, as the ad suggests.
$1.5 billion in Taxes and Fees?
As regular readers of FactCheck.org know by now, although Romney opposed across-the-board tax increases, he did balance the state budget by raising a number of fees and closing some corporate tax loopholes. The question is whether it added up to $1.5 billion, as the ad claims.
As we have reported in the past, there is some disagreement over the amount of the annual revenues raised by those fees and the closing of corporate tax loopholes. The Massachusetts Department of Administration and Finance put the fee total at $260 million a year and the corporate tax change at $174 million a year, and the independent Massachusetts Taxpayers Foundation said both fees and taxes totaled $740 million to $750 million a year (split about evenly between fees and corporate taxes).
So how does the ad get to $1.5 billion?
The Obama campaign says that it's a four-year figure — something not obvious to those watching the ad. And in its backup material, the campaign cites media estimates of the four-year total for only corporate tax loophole closures (excluding fees — which is the focus of the ad).
Michael Widmer of the Massachusetts Taxpayers Foundation said its policy is always to cite the annual impact of taxes and fees. You could look at the 10-year, 20-year or even the 50-year impact, he said, if you really wanted to make the figure look huge. But where do you stop?
The Obama campaign could truthfully have said the total for both fees and loophole closings was "as high as $3 billion over four years." But for some reason, it picked a figure that included only loophole closures — and failed to make clear this wasn't an annual figure.
The ad also claims Romney hiked over 1,000 fees. It then begins to list a number of them. But when we checked the backup material provided by the Obama campaign, some of the fees weren't as far-reaching as the ad might lead a viewer to believe.
For example, the ad says Romney hiked fees:
On driver's licenses. The backup material cites increased fees for a Class A driver's license, from $52.50 to $60; Class B, from $40 to $50; and Class C and Class M, from $33.75 to $40. But according to the Massachusetts Department of Transportation website, those are licenses for large commercial vehicles — tractor trailers, buses and the like — as well as motorcycles. The fee hikes were not for driver's licenses to operate a car.
On fishermen. The ad mentions fishermen in the same breath as gun owners. And while it's true that the fee increases directly apply to any recreational hunter or gun owner, the same cannot be said for the fees that apply to fishermen. The fees applying to fishermen include those for a "Master Digger" shellfish license (for residents as well as non-residents), and a $6 charge for each half-bushel of shellfish depuration (toxin-removal). These are fees that wouldn't apply to the average recreational fisherman in Massachusetts.
On school bus rides. This one is a real stretch. The state doesn't impose fees for school bus service. Bus service is provided by the school districts. Rather, Romney created new powers for municipal school committees to assess students a transportation fee to cover the cost incurred by the district. Michael Gilbert, with the Massachusetts Association of School Committees, argues that Romney indirectly hiked these fees because he slashed state transportation reimbursements to local and regional districts. Local districts weren't required to enact bus fees, but many of them had to in order to balance their constrained budgets, he said.
On milk. Although the ad pictures a girl drinking a glass of milk, Romney didn't place a fee on the purchase of a gallon of milk from your local grocery store. Rather, as the backup material notes, Romney increased the fee for a milk dealer license from $5 to $10; increased the permit fee for a pasteurization plant from $50 to $100; and increased the fee to operate a device for the determination of butterfat content of milk from $10 to $20. Perhaps those costs are passed on to the consumer, but the effect on any single purchase would be infinitesimal.
On mental health services. The fees actually only applied to mental health counselor applications, registrations and renewals. Increases ranging from $32 to $45 in the cost of maintaining a license would be almost impossible to spot in the price of mental health services for patients, whom the ad suggests would be affected.
More broadly, the ad is simply incorrect when it says that Romney raised taxes and fees on "everyone else."
In a debate during his first run for president in 2008, Romney defended the fees, saying they were necessary to help close a $3 billion budget shortfall, and because "we found that some fees hadn't been raised in as many as 20 years."
Specifically, Romney said, "these were not broad-based fees for things like getting your driver's license or your license plate for your car, but instead something like the cost of a sign on the interstate and how much it was going to cost to publish a McDonald's or a Burger King sign on the interstate. We went from, like, $200 a sign to $2,000 a sign to raise money for our state in a way that was consistent with the what the market had done over the ensuing years."
Michael Widmer of the Massachusetts Taxpayers Foundation told us some of the fees were more broad-based than Romney let on — such as a fee on deeds for real estate transactions (the largest revenue producer). That fee hike hit anyone buying or selling real estate in a given year, but it didn't hit "everyone."
As for the fees being outdated, that may have been the case for some fees, Widmer said, but there was never any analysis presented to justify the fee increases. Rather, he said, it was clear the overriding purpose was to raise revenue to help close the budget gap.
One last note on the ad's contention that Romney raised taxes and fees on "everyone else" but millionaires. In 2004, 2005 and 2006, Romney proposed cutting the state's income tax rate from 5.3 percent to 5 percent — which would have provided a bit of tax relief to everyone, including the middle class. In every case, his efforts were rebuffed by the Democratic Legislature.
Widmer notes that although Romney supported those tax cuts — which would have slashed state revenues by about $750 million — he never described a plan to reduce spending in a way that would make the cuts possible.
"From a fiscal viewpoint, it was not a fully developed proposal," Widmer said.
— Robert Farley and Nathan Emmons
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For those of you who have followed my historical blog, ArkieGamer, you may not believe this, but I actually finished all of the miniatures that I had ordered, BEFORE placing an order for more minis. Or at least before they arrived.
Without further ado, here are my Khurasan Federals.
A typical squad, accompanied by a Polecat armored car.
I'll probably use these guys as a recon element, of some sort.
Probably the least useful unit I bought, as far as getting it on the table. On-board artillery, even at a small scale like 15mm, doesn't make much sense for close-in skirmish.
I love this little scout drone. I thought about basing him with acrylic rod, but decided that black is best.
Probably the least useful unit I bought, as far as getting it on the table. On-board artillery, even at this level, doesn't make much sense for close-in skirmish.
I tend to get obsessed about things and do extensive amounts of web research until I begin to grasp the zeitgeist of whatever said thing might happen to be. I'm sure I'm not the only one. Before I begin to digress too badly about being an obsessive nerd, I'll just say that one of the models that particularly struck my fancy when I was relentlessly searching for information on 15mm sci-fi wargaming was Khurasan Miniature's Polecat.
The Polecat is basically a futuristic HMMWV with the sort of functionalist industrial aesthetic that James Cameron (or Ridley Scott, for that matter) captured in Aliens. I really love it. In fact, it's the model that pushed me over the edge into beginning to collect, paint, and hopefully game 15mm science fiction.
Like the Humvee, the Polecat is ubiquitous. They can even be found at battalion HQ.
Here's my version. I used a similar palette to my infantry, with Vallejo Russian uniform being the primary ingredient. I went a little overboard on the weathering, and will probably try to tone it down a notch for my next 15mm armored vehicle. The "4" is supposed to resemble the Vilani font from good old Traveller, but it was quite difficult to execute at this scale!
If When I do another of these, it will have to have been nicknamed and tagged as 'Pepe' by its crew.
The Polecat is ubiquitous, like the Humvee. They can even be found at battalion HQ.
The Polecat carries four, according to Khurasan's website fluff. I guess the last guy is manning the turret on overwatch.
I really muffed the Vilani style "4" here. The bottom bar should be much thinner. The tail lights look much better at table distance than they do in this photo. They're just vermilion, some sort of GW bright yellow, and a medium blue with Future Wax varnished over.
I've painted my first 15mm Sci Fi figures, as personal first for the scale and the genre. These are Khurasan Miniatures Federal troops. I have another 9 figures to paint, which will give me one squad, a command group, and a couple of heavy weapon attachments. I've promised myself I won't be ordering any new figures until I have the current batch finished, but the ultimate goal is to build up a platoon and a couple of light armor vehicles.
Obviously I have a long way to go, but 15mm is a blast to paint.Laying down the base colors in 15mm is very quick, and this allows you spend more time on the fun stuff.
Primary colors on these guys are Vallejo Russian Uniform. The hardware is Russian Uniform mixed with German Camo Dark Green. I started out with white primer and did a couple of black washes along the way. I think my paint is way too thick, and I'd really like to get the armor looking like some sort of satiny composite plastic rather than the painted gypsum board that it now resembles.
They're based on US dimes. I'm in the process of deciding whether or not to color code the bases for unit type/number.
Obviously I have a long way to go, but 15mm is a blast to paint. It's quick to lay down the basic colors in 15mm, which lets you spend more time on the fun stuff.
From left to right: Trooper with grenade launcher, trooper with SAW, and what is probably a point man giving a hand signal.


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Health evaluates asking for negative PCR test again to enter the country
Health Surveillance Officials are analyzing once again the tequirement was eliminated last October
HealthRedaqted
QCOSTARICA – The Health Surveillance Directorate of the Ministry of Health is evaluating the possibility of again requiring a negative PCR covid-19 test to travelers who enter Costa Rica.
This measure had been eliminated on October 22.
The possibility of reinstating the provision was recognized this Wednesday morning by the interim Minister of Health, Pedro González, during a press conference.
"This specific requirement is under analysis by the technical groups. It was removed at the time because the negative test, due to the characteristics of the test and the disease, did not give certainty that the traveler did not have the disease, could be asymptomatic or in the incubation period, and this would possibly not be reflected in the result of the test," said González.
"At this moment it is being analyzed if these requirements are to be put back in," he added.
The interim Minister did not specify when the analysis would be completed and when if the measure would be reinstated.
The United States announced this Tuesday that it will request the negative test from all those who enter their country by air as of January 26.
Canada also requires as of January 7 a negative PCR test for all travelers to that country.
PAHO with caution
The Pan American Health Organization (PAHO) gave its opinion of requesting negative tests for entry to any country during its press conference this Wednesday morning.
"PAHO does not support the use of tests as tools to mitigate the spread of covid-19," said Ciro Ugarte, PAHO's Director of Emergencies.
The specialist pointed out that this is due to several reasons. The first is that not all tests have the same level of confidence. Furthermore, this creates a sense of false security.
"The result of the test takes into account the moment in which it was taken, but the person can be infected at any time between the sample was taken and the trip began," exemplified Ugarte.
For the specialist, another reason is that this forces countries to have a system that ensures their travelers that they have tests available to take before traveling.
He added that rapid antigen tests should prioritize efforts to search for cases within the country and not for travelers.
Ugarte stated that there are other measures that have more impact on public health, such as performing a PCR test upon arrival in the destination country.
Health Surveillance
Pedro González
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Visitors from China, Hong Kong, Taiwan and Macao must have a negative Covid-19 test
QCOSTARICA - The Ministry of Health has announced that, as of...

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Authors & Presenters
Bill Hall
Bill Hall is the former director of educational leadership and professional development for Brevard Public Schools in Florida. He has served as a classroom teacher, assistant principal, and principal.
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Bill Hall is the former director of educational leadership and professional development for Brevard Public Schools in Florida. He has served as a classroom teacher, assistant principal, and principal. As a teacher, Bill was recognized as Teacher of the Year at Sherwood Elementary School in Melbourne, Florida. As the principal of Surfside Elementary School in Satellite Beach, he received the key to the city in recognition of his outstanding leadership and service to the students and community. During the last half of his career, Bill served as the director of educational leadership and professional development for the district.
As director of educational leadership, his responsibilities included leading the district's leadership development initiatives, the principal and assistant principal preparation programs, and the implementation of Professional Learning Communities at Work® in support of the district's instructional model. Under Bill's leadership, 27 of Brevard County's 85 schools were recognized as Model PLC at Work Schools by Solution Tree.
He is a past president of the Florida Association for Staff Development (FASD), now Learning Forward Florida. He is also a senior fellow of the Institute for Development of Educational Activities, or /I/D/E/A/, Academy of Fellows.
For his years of service to FASD, he received its Distinguished Contribution Award. For two consecutive years, he received the Seiji Horiuchi Memorial Award from the US Jaycees in recognition of his leadership as Florida's state individual development chairman.
Bill is the author of Powerful Guiding Coalitions: How to Build and Sustain the Leadership Team in Your PLC at Work®, published by Solution Tree Press. His articles on leadership development have been published in AllThingsPLC Magazine, the Journal of Staff Development (JSD), the National Forum of Educational Administration and Supervision Journal, and the District Management Journal (DMJ).
Bill received his bachelor's in English from Erskine College and his master's in administration and supervision from Nova Southeastern University. He is certified in school principalship and elementary education.
PLC at Work® experts have proven success in leading the PLC at Work process in a school or district. Work with them to implement a focus on learning, build a collaborative culture, and create a results orientation that leads to sustained, substantive school improvement.
Presentation Titles
Presentations by Bill Hall
What PLC Schools Do Differently: Seven Strategies for Success
Becoming a Professional Learning Community: A Journey, Not a Destination
The Framework for Developing Teacher Leadership through Professional Learning Communities
Leading Quality Professional Learning Communities
Districtwide Implementation of Professional Learning Communities
Ensuring Sustainability through Effective Succession Planning
The Six Essential Strategies for Ensuring Leadership Development and Leadership Sustainability
Brevard Public Schools: The Journey to Becoming a Role Model for Organizational Performance Excellence
Ensuring Quality Leadership
Leadership Development: The Critical Element in Sustaining the Cultural Changes of a Professional Learning Community
Succession Planning and Management in Public Education
Dispersing Leadership and Planning for Succession
The Sterling Process/APQC
Restructuring of School Leadership Development Programs
Florida's Professional Development System Evaluation Protocol
Leadership for School Leaders
Leaving a Legacy of Leadership in Your Organization
Overview of Professional Learning Communities
Sharpening the Point: Professional Development Academy
Performance Appraisal Systems
Professional Learning Communities at Work®
Teacher Leadership: Unleashing the Power of Potential
Hot Topics: Professional Learning Communities
"Bill was very patient with my staff. His presentation was fantastic."
Kathy Briggs, principal, Riverbank Unified School District, California
"LOVE Bill Hall. He is an amazing presenter and consummate professional."
Ian Squibb, principal, Fremont Unified School District, California
Powerful Guiding Coalitions


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Centre "Playing Politics" Over Releasing Rajiv Gandhi Assassins: Congress
HomeAll IndiaCentre "Playing Politics" Over Releasing Rajiv Gandhi Assassins: Congress
Seven people are serving life sentence for their role in the assassination of former prime minister Rajiv Gandhi by a suicide bomber.
All IndiaPress Trust of IndiaUpdated: September 10, 2018 10:58 pm IST
Rajiv Gandhi was assassinated on May 21, 1991.
The Congress Monday accused the Centre and the Tamil Nadu government of "playing politics" over a cabinet recommendation to release all seven life convicts in the Rajiv Gandhi assassination case, adding that the State cannot derelict its duty in punishing terrorists.
Congress chief spokesperson Randeep Surjewala said there should be no compromise on terrorism and terrorists and it was time to show the mirror to the PM Modi and All India Anna Dravida Munnetra Kazhagam (AIADMK) governments who were adopting a "soft approach" towards them (terrorists).
"Bereft of who says what, as a citizen and a Congressman, we are of a clear opinion that the State cannot derelict its duty of punishing terrorists and terrorism.
"Those who swear by pseudo nationalism day in and day out are today doing politics over assassins of Rajiv Gandhi Ji and dozens of others who were killed by these terrorists, who have been punished by the Supreme Court, and never been granted any pardon," he said.
"It is time to show the mirror of truth to Prime Minister Narendra Modi and his Government who are shielding terrorism and terrorists by such acts," Mr Surjewala said.
"We ask a simple question, bereft of what th large-hearted (then) Congress President Sonia Gandhi or the current Congress President Rahul Gandhi or his sister Priyanka Gandhi have said viz-a-viz having no enmity towards the assassins of Rajiv Ji. Does the State have a duty to protect terrorists or to find terrorists?
"Is it not true that BJP's ally AIADMK and BJP-appointed Governor are proceeding to recommend, reprieve, remission and release of assassins and terrorists? Is that the policy of the State now? Are we going to release assassins of the former PM and innocent civilians and policemen accused of terrorism?," the Congress leader said.
The AIADMK government in Tamil Nadu Sunday recommended to Governor Banwarilal Purohit the release of all seven life convicts in the Rajiv Gandhi assassination case, a move hailed by most political parties in the state barring the Congress.
A resolution to this effect was adopted by the state cabinet which met under Chief Minister K Palaniswami's leadership.
The cabinet decided to release the convicts V Sriharan alias Murugan, T Suthanthiraraja alias Santhan, A G Perarivalan alias Arivu, Jayakumar, Robert Payas and Nalini, under Article 161 of the Constitution, which relates to the power of the Governor to grant pardons and to suspend, remit and commute sentences in certain cases.
The seven are serving life sentence for their role in the assassination of former prime minister Rajiv Gandhi by an Liberation Tigers of Tamil Eelam (LTTE) suicide bomber on May 21, 1991 in Sriperumpudur.
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Join WTS-LA on June 21 to find out how transportation agencies will prioritize that need to keep our infrastructure safe and sound.
People love new infrastructure: A new bridge. A new highway. A new rail line. All provide great optics for ribbon-cuttings and speeches. But keeping existing infrastructure in a state of good repair is equally important, in some cases more so. Yet existing infrastructure often gets little attention and less funding. WTS-LA knows that's wrong. And so do many transportation agencies in Southern California. That's why they work tirelessly to put their money where their existing infrastructure is, in addition to developing new projects. And on June 21 in an afternoon panel discussion, WTS-LA welcomes key transportation players that are fighting the good fight to keep Southern California infrastructure in a state of good repair.
Maintenance and repair may not make for good optics. But those projects are essential to improving mobility in Southern California. And there is no shortage of need when it comes to infrastructure repair. Join WTS-LA on June 21 to find out how transportation agencies will prioritize that need to keep our infrastructure safe and sound.
Lunch: 12:00 - 1:30 p.m.
Cancellations must be made prior to Friday, June 16 at 5 p.m. to receive a full refund. For more information about this event or WTS-LA, visit our website.


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Home/Headlines/This is the face of poverty in Nigeria
HeadlinesOpinion
This is the face of poverty in Nigeria
Poverty is often presented as statistics. But what the Yusufu Bala Usman Institute did on September 21, 2022 was to produce a book, The Face of Poverty in Nigeria, which focused on the faces behind the statistics.
As one of the reviewers, I summarised the Nigerian situation thus: If you are poor in thinking, you are poor. If others think for you, you are poor. If you follow other people's culture, you are poor. If foreigners decide your beliefs, you are poor. If others decide your economic policies and programmes, you cannot but be dirt poor.
My argument is that poverty, which is the severe deprivation of a person resulting in his inability to meet the basic needs of life, manifests in various ways, situations and circumstances.
In unveiling the actual faces of the victims, the publication reminds us that Nigeria is nesting on top of a volcano. If, like the research reveals: "Over 90 million people in Nigeria, (or) nearly 50 per cent of its population, are living in poverty", then the volcano on which we sit, cannot but be an active one. That figure of over 90 million Nigerians in poverty, is the combined population of over 15 European countries, including Switzerland, Bulgaria, Norway, Denmark, Finland, Slovakia and Ireland.
The poverty statistics is far grimmer if we realise that the book adopts as poor, anyone earning less than N377 per day rather than the World Bank's $2.15 per day.
Employment in a capitalist society, in many cases, determines whether a person lives in poverty or not. The Report states that 13.1 million are unemployed, while 11.3 million are underemployed. In a sense, this 24 million Nigerians are already sentenced to purgatory. More so when there is no unemployment benefit in the country.
But even this class, has sub-classes and a gender dimension. The book states that while only 20 million of the total employed population of 46 million are women, less than a million women are in formal employment, while 95 per cent are engaged in the vagaries of the informal sector.
It identifies one of the sub-classes as the dispossessed citizens who have weak access to good-quality public goods (infrastructure) and services, and limited voice in public policy making.
If I may add, in most cases, the dispossessed have no hope in the future and some of them have an uncommon sense of duty. Let me give an example. In the suburbs of Abuja, some of them dig up portions of the tarred road, purport to be engaged in community service by partially filling the resultant potholes with sand and then ask motorists who are forced to slow down at such places to give them alms to 'appreciate' their efforts at road repair.
Yet, the rural poor tend to be in a worse condition. The publication indicates that 52.1 per cent of rural dwellers are poor compared to 18.04 per cent of urban dwellers.
My conclusion is that there is worse poverty in the rural areas because the ruling elites concentrate infrastructure like health centres, schools and roads in the urban centres and the minimum living level, is lower in the rural areas.
There is also a regional dimension. The book states that nine of the 10 poorest states in the country are in the North. Some of the reasons it advanced for this situation include terrorism, banditry, kidnapping, and pastoralists and herders' violence. But I think to this should be added the issue of Western education; the pattern is: the less educated a place is, the greater the level of poverty.
The region is also negatively impacted by desertification as 50-75 per cent of 10 of the 19 Northern states may be experiencing desertification.
Employment is central in combating poverty. However, I must point out that employment is not an automatic gate pass out of poverty. First, wages can be very low, which is a main reason why lecturers have shut down the public universities in the country for over seven months. The second is non-payment of salaries.
Mass retrenchment is identified as one of the worst things that can happen to workers because it translates to automatic stoppage of wages and drop in living standards. This is exacerbated by the privatisation of public-owned companies.
The book also highlights the challenges faced by the pensioner as going on pension is like being sentenced to economic death. Given this reality, it is not uncommon for workers to doctor their ages in order to remain in employment for as long as possible or until death do them part.
The research concluded that those in wage employment experience severe shock when their income is suddenly cut off, whether"by retirement, illness, or retrenchment. Its findings are that there is so much poverty in the country mainly due to the neo-liberal economic policies which includes reduction in social spending on programmes like healthcare, education, privatisation and mass retrenchment.
These, it reports, results in lack of mass access to basic needs like electricity to which only 22 per cent of the poorest households have access; education with 10.5 million children out of school, healthcare with a 1.95 to 1,000 ratio of nurses, midwives and doctors, to patients. There are also issues of population control.
The book concludes that the poverty-inducing policies of the Western internal financial institutions are a major cause of mass poverty in Nigeria.
There are, of course, the added challenges of currency devaluation, primitive accumulation, and the very logic of the capitalist system which is based on exploitation.
A very important contribution of the book is how irrigation can be a primary way to increase the incomes of farmers and pastoralists, and additionally, reduce their clashes.
In some instances, the 189-page book leaves gaps which may give room for some conclusions as to motives. For instance, in many cases of land grab, while the herders were extensively interviewed, the farmers living in the same locality, were not interviewed. I think the researchers need to interview both, or all sides of an issue, and not accept a given narrative as nothing but the truth.
The book, including its suggestions on alternative economic policies, checking land grabbing, adopting irrigation and placing humans in the centre of statistics, is a very important addition to the study of poverty in Nigeria. I would also suggest some background research on poverty in pre-colonial and colonial times. For example, was homelessness possible or rampant before colonialism? Except for such challenges as climate and ecological challenges, was food self-sufficiency an issue before the imposition of cash crops?
The book, supported by the Rosa Luxemburg Foundation, is structured and written in simple language that can be understood by the average newspaper reader. So, I submit that in terms of communication, it succeeds.
Lakemfa Nigeria Poverty Rosa Luxemburg Foundation
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HUGH Jackman has made the million dreams of his Aussie fans come true, confirming he will bring his one-man show home.
Announced in true showbiz style by Jackman himself at the Museum of Contemporary Art in Sydney's The Rocks this morning, Hugh Jackman The Show will hit Australian stages in August.
It will feature songs from The Greatest Showman, Lea Misérables and other Broadway hits, accompanied by a live orchestra.
The 50-year-old Academy Award-nominated, Golden Globe and Tony Award-winning triple threat said: "I'm my happiest when I'm on stage."
Prior to the big announcement, the Australian Girls Choir opened formalities with a rendition of Million Dreams, and joined my Jackman halfway through.
Jackman then welcomed special guest, and good friend, Keala Settle - who played The Bearded Lady in The Greatest Showman - treated media to a live, goosebump-inducing performance of the film's mega hit This Is Me.
Settle will also join Jackman for the Australian leg of his World Tour, which includes two shows at the Brisbane Entertainment Centre in August.
Jackman's "very good friend," Andy Lee also made an appearance to host a musical theatre inspired question and answer session, where Lee sung his questions to Jackman and Settle, who sung their answers back.
Tickets go on sale on March 7 at 9.30am AEDT via Ticketek.


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Greenvale is a residential suburb on the outskirts of metropolitan Melbourne. It is situated between Bulla and Craigieburn and is 23 km north-west of central Melbourne.
On its west there is the Moonee Ponds Creek, flowing in an arc around the historic Woodlands Estate (1843) on Gellibrand Hill. The centrepiece is the heritage listed Woodlands homestead (1843), set in Woodlands Historic Park. The park includes heritage listed farmhouse ruins and a Koorie cemetery. The Yuroke Creek and several tributary headwaters flow through Greenvale and enter the Moonee Ponds Creek further south at Broadmeadows. Immediately north of the Woodlands Estate is Oaklands Junction, home of the Oaklands Hunt Club. The area has several stud properties and a large quarry.
The Woodlands site is next to the former Greenvale elderly persons' hospital, originally the Greenvale Sanitarium for Consumptives (1905). The site was chosen for its elevation above sea level, being described as wooded uplands. Consisting of 121 hectares it had landscaped grounds, a fruit and vegetable garden, and was intended mainly for patients in reduced circumstances. Among the numerous buildings on the site there is the modernist ward and hall (1950) designed by the noted architect, Percy Everett.
Greenvale's better quality land attracted sufficient settlers for a school to be opened in 1868. The school, along with a Primitive Methodist church (1869) and recreation reserve, constituted the district's centre. The area was a rural community until the late 1980s, although the building of a domestic water reservoir on the Yuroke Creek's tributaries in 1971 signalled the approach of suburbanisation.
Most of Greenvale between Woodlands Historic Park and the reservoir is subdivided for residential living. Subdivisions began in 1972. A larger primary school (1988) is more centrally located than the original site which became a Seventh Day Adventist church. There is a shopping centre and numerous reserves, several along drainage reserves. There is also a park on the edge of the reservoir.
Greenvale's housing blocks were mostly 0.5 to 5 hectares, and the houses correspondingly large. There is a wide range of styles, from Tudor to Californian bungalow. One house has 30 Grecian columns. After the initial appeal of large blocks, smaller ones of 0.2 hectares have had an upwards price movement.
Greenvale has three distinct living areas. The early village, including Methodist church and recreation area, mostly rural, is between Woodlands and Mickleham Road. The first modern suburb, with large house blocks, is east of Mickleham Road. Thirdly, there is the smaller-block suburb adjacent to the reservoir, north of Somerton Road. Some housing development was delayed in 2010 after the discovery of the critically endangered golden sun moth.
Greenvale Village shopping centre (1988) and the school (664 pupils, 2014) are in the first suburb. Kolbe Catholic college (2006) is in the smaller-block suburb, sharing its catchment with Roxburgh Park. This suburb, still in the course of development, is on the slopes of Aitken Hill (Mount Yuroke), a volcanic cone of elevation 265 metres. The cone, named after the pioneer pastoralist John Aitken, is flanked by a public parkland. The area around Aitken Hill is also known as Greenvale Lakes, which merges north to the suburb of Craigieburn.
Greenvale has a significant number of residents of Italian origin. At the 2011 census 25.1% claimed Italian ancestry.


text:
package org.javalite.activejdbc;

import java.io.Serializable;

public class ColumnMetadata implements Serializable {

    private final String columnName;
    private final String typeName;
    private final int columnSize;

    public ColumnMetadata(String columnName, String  typeName, int columnSize) {
        this.columnName = columnName;
        this.typeName = typeName;
        this.columnSize = columnSize;
    }

    /**
     * Column name as reported by DBMS driver.
     * @return column name as reported by DBMS driver.
     */
    public String getColumnName() {
        return columnName;
    }

    /**
     * Column size as reported by DBMS driver.
     *
     * @return column size as reported by DBMS driver.
     */
    public int getColumnSize() {
        return columnSize;
    }

    /**
     * Column type name as reported by DBMS driver.
     *
     * @return column type name as reported by DBMS driver.
     */
    public String getTypeName() {
        return typeName;
    }

    @Override
    public String toString() {
        return "[ columnName=" + columnName
                + ", typeName=" + typeName
                + ", columnSize=" + columnSize
                + "]";
    }
}

redpajama_set_name: RedPajamaGithub


text:
\section{Introduction}

A class of codes is said to be asymptotically good
if there is a sequence $C_1,C_2,\cdots$ of codes in the class with length
$n_i$ of $C_i$ going to infinity such that
 both the rate of $C_i$ and the relative minimum distance of
$C_i$ are positively bounded from below.
By a Varshamov's probabilistic argument \cite{V}, linear codes
over a finite field are asymptotically good.

Cyclic codes are an important class of codes and studied extensively.
It is a long-standing open question:
are the cyclic codes asymptotically good?
The question was first asked
by Assmus, Mattson, and Turyn in 1966 (see \cite{AMT}).
Many related researches are unfolded by
revolving around the question, for example, studies on asymptotic properties
of quasi-cyclic codes, group codes, and so on. A classical result is that
the quasi-cyclic codes of index $2$ are asymptotically good (see \cite{CPW}, \cite{C}, \cite{K}).
Note that quasi-cyclic codes of index $1$ are just cyclic codes.
Recently, \cite{FL16} introduced quasi-cyclic codes of fractional index,
and proved that the quasi-cyclic codes of fractional index
between $1$ and $2$ are asymptotically good.

In 1994, Hammons et al. discovered that some
good non-linear binary codes can be viewed as the Gray map
images of some linear codes over $\mathbb{Z}_4$ (see \cite{HKCSS}).
From then on, the study of codes over
$\mathbb{Z}_4$ and other finite rings has been developing.
In particular, linear and cyclic codes over $\mathbb{Z}_4$
have been studied
 intensively (\cite{AS}, \cite{AGO}, \cite{AO}, \cite{PQ}).

In 1973, Delsarte \cite{D5}) defined additive codes
as subgroups of the underlying abelian groups in translation association schemes.
For the binary Hamming scheme, i.e., when the underlying abelian group
is of order $2^n$, only the abelian groups of the form
$\mathbb{Z}_2^\alpha\times\mathbb{Z}_4^\beta$
with $\alpha+2\beta=n$ appear.
The subgroups $C$ of $\mathbb{Z}_2^\alpha\times\mathbb{Z}_4^\beta$
are called $\mathbb{Z}_2\mathbb{Z}_4$-additive
codes (\cite{BBDF}, \cite{BFPRV}, \cite{RRR}).
If $\beta=0$, then $\mathbb{Z}_2\mathbb{Z}_4$-additive
codes are binary linear codes.
If $\alpha=0$, then $\mathbb{Z}_2\mathbb{Z}_4$-additive
codes are the quaternary linear codes over $\mathbb{Z}_4$.  So,
this class of codes contains all binary and quaternary linear codes as a subclass.
Moreover, this class of codes has shown at its early stage
some promising applications. For example,
perfect $\mathbb{Z}_2\mathbb{Z}_4$-additive codes
have been utilized in the subject of steganography (\cite{RRR}).
Therefore,
$\mathbb{Z}_2\mathbb{Z}_4$-additive codes are an important class of codes.
In recent years, the structure and properties of $\mathbb{Z}_2\mathbb{Z}_4$-additive codes have been intensely studied (see \cite{BFPRV}).
In \cite{ASA}, Abualrub, Siap and Aydin first introduced $\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic codes.
\cite{BFT} studied the generator matrices and dual codes of $\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic codes.
Borges, Dougherty, Fern\'andez-C\'ordoba and Ten-Valls study the rank and the kernel of $\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic codes in \cite{BDFT}.

An interesting question we are concerned with comes up:
are the  $\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic codes
asymptotically good?

In this paper we answer the question positively.
A key innovation to solve the question is that: within
$\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic codes
we find a class of codes such that
the codes in the class possess good algebraic structures,  in fact,
they are algebraically isomorphic to the binary quasi-cyclic codes
of index two generated by pairs of polynomials; and,
though the algebraic isomorphism dos not preserve
the rates and the relative minimum distances of the codes,
we have a technical probabilistic method
to determine asymptotically the rates and the relative minimum distances
of the codes in the class.

In Section 2, we state some preliminaries on
$\mathbb{Z}_2\mathbb{Z}_4$-additive codes and
$\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic codes.
The rate $R(C)$ and the relative minimum distance
$\Delta(C)$ of a $\mathbb{Z}_2\mathbb{Z}_4$-additive code $C$
are also defined in this section.

In Section 3, we construct the class of the
$\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic codes $C_{a,b}$ (defined below)
which will play an important role as mentioned above,
By $R_m=\mathbb{Z}_2[X]/\langle X^{m}-1\rangle$
we denote the residue ring of the polynomial ring $\mathbb{Z}_2[X]$
over $\mathbb{Z}_2$ modulo the ideal $\langle X^{m}-1\rangle$
generated by $X^{m}-1$.
And $R'_{m}=\mathbb{Z}_4[X]/\langle X^{m}-1\rangle$ similarly.
We define the {\em $\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic codes
with cycle length $m$} to be the $R'_m$-submodules of $ R_m\times R'_{m}$.
With $\mathbb{Z}_2\mathbb{Z}_4$ as an alphabet,
such codes has many properties similar to the classical cyclic codes,
e.g., they can be implemented by feed-back shift registers.
For our purpose, we are interested in
the $R'_m$-submodules of $ R_m\times 2R'_{m}$
where $2R'_{m}$ is the ideal of $R'_m$ generated by $2$.
Furthermore, to ignore the trivial case, we consider the ideal
$J_m=\langle X-1\rangle_{R_m}$ of $R_m$ generated by $X-1$,
and the ideal $2J'_m=\langle 2(X-1)\rangle_{R'_m}$
of $R'_m$ generated by $2(X-1)$. Then, for any polynomials $a(X),b(X)\in J_m$,
we have an $R'_m$-submodule $C_{a,b}\subseteq J_m\times 2J'_m$
generated by $(a(X),2b(X))$ in a natural way:
$C_{a,b}=\big\{\big(c(X)a(X),2c(X)b(X)\big)\,\big|\, c(X)\in J_m\big\}$.
By investigating  the two circulant matrices associated
with the two polynomials $a(X)$ and $b(X)$,
we get the generator matrix of $C_{a,b}$ from the
polynomial pair $a(X),b(X)$. And we obtain necessary information
about the number of the ideals of $J_m$ and their dimensions
for later using to estimate the relative minimum
weight $\Delta(C_{a,b})$ and the rate $R(C_{a,b})$.

In Section 4, we view $J_{m}\times 2J'_m$ as a probability space with
equal probability distribution, so that the
$\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic codes $C_{a,b}$
with cycle length $m$ for $a(X),b(X)\in J_{m}$
is a random code, hence the relative minimum distance $\Delta(C_{a,b})$
and the rate $R(C_{a,b})$ are random variables.
We study the asymptotic properties with $m\to\infty$
of the two random variables.
Given a real number $\delta$ such that $0<\delta<1/3$
and the binary entropy $H(3\delta/2)<1/2$
(it is about that $0<\delta\le 0.0733$).
For each $c(X)\in J_m$, the word $\big(c(X)a(X),2c(X)b(X)\big)\in C_{a,b}$
is a random codeword which gives a Bernoulli variable $Y_c$:
$Y_c=1$ if the random codeword is a non-zero one with relative weight $ \le\delta$,
and $Y_c=0$ otherwise.
Applying \cite[Theorem 3.3]{FL} which estimates the number
of the codewords in a code whose relative weights are bounded from above,
we calculate the expectation of $Y_c$;  and then we prove that
the probability that $\Delta(C_{a,b})>\delta$ is convergent to $1$.
Next, using an asymptotic property of the dimensions
of irreducible modules in $J_m$, we prove that, in an asymptotic sense,
the probability that $R(C_{a,b})=\frac{1}{3}-\frac{1}{3m}$
is also convergent to $1$.
Finally, it is an obvious consequence that
$\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic codes are asymptotically good.

\section{Preliminaries}
First of all, we sketch briefly
{\em $\mathbb{Z}_2\mathbb{Z}_4$-additive codes $C$}.
Any subgroup $C$ of $\mathbb{Z}_2^{\alpha}\times\mathbb{Z}_4^{\beta}$
is called a $\mathbb{Z}_2\mathbb{Z}_4$-additive code.
It is obvious that $\mathbb{Z}_2^{\alpha}\times\mathbb{Z}_4^{\beta}$
is a $\mathbb{Z}_4$-module, and its subgroups are $\mathbb{Z}_4$-submodules.

As usual, the weight of
${\bf a}=(a,a')\in\mathbb{Z}_2^{\alpha}\times\mathbb{Z}_4^{\beta}$
is defined by
\begin{equation}\label{weight}
 {\rm wt}({\bf a})={\rm wt}(a,a')={\rm wt}_H(a)+{\rm wt}_L(a'),
\end{equation}
where ${\rm wt}_H(a)$ is the Hamming weight of $a$
and ${\rm wt}_L(a')$ is the Lee weight of~$a'$.
The minimum weight of the $\mathbb{Z}_2\mathbb{Z}_4$-additive code $C$
is defined as
${\rm wt}(C)=\min\big\{{\rm wt}({\bf c})\,\big|\,{\bf c}\in C \backslash\{{\bf 0}\}\big\}$.
For the well-known Gray map $\phi: \mathbb{Z}_4\to\mathbb{Z}_2^2$,
$\phi(0)=(0,0)$, $\phi(1)=(0,1)$, $\phi(2)=(1,1)$, and $\phi(3)=(1,0)$,
there is a natural extension:
\begin{equation}
\begin{array}{cccl}
\Phi:& \mathbb{Z}_2^{\alpha}\times\mathbb{Z}_4^{\beta}
  &\to&\quad\mathbb{Z}_2^{n}, \\
 & (a, a') &\mapsto& \Phi(a, a')=\big(a_0,\cdots,a_{\alpha-1},
  \phi(a'_0), \cdots, \phi(a'_{\beta-1})\big);
\end{array}
\end{equation}
where $n=\alpha+2 \beta$,
$a=(a_0,\cdots,a_{\alpha-1})\in\mathbb{Z}_2^{\alpha}$ and
$a'=(a'_0,\cdots,a'_{\beta-1})\in\mathbb{Z}_4^{\beta}$.
The map $\Phi$ is an isometry which transforms
the weight in Eqn \eqref{weight}
to the Hamming weigh in $\mathbb{Z}_2^{n}$.
We call the fraction
\begin{equation}\label{def relat dis}
 \Delta(C)=\frac{{\rm wt}(C)}{n}
\end{equation}
the {\em relative minimum distance of $C$}.

Since $C$ is a subgroup of $\mathbb{Z}_2^{\alpha}\times\mathbb{Z}_4^{\beta}$,
it is also isomorphic to an abelian structure like
$\mathbb{Z}_2^{\gamma}\times\mathbb{Z}_4^{\delta}$.
Therefore, $C$ is of type $2^{\gamma}4^{\delta}$ as a group,
the number of elements of $C$ is $|C|=2^{\gamma+2\delta}$,
and the number of the elements of order $2$ in $C$
equals to $2^{\gamma+\delta}$.
We say that the fraction
\begin{equation}\label{def rate}
 R(C)=\frac{\gamma+2\delta}{n}
\end{equation}
is the {\em rate} of the code $C$.

\smallskip
There is an additive group homomorphism
from $\mathbb{Z}_2$ into $\mathbb{Z}_4$:
\begin{equation}\label{2-map}
 \mathbb{Z}_2\to\mathbb{Z}_4,~~ a\mapsto 2a,
\end{equation}
and the image of the homomorphism is
$2\mathbb{Z}_4=\{0,2\}\subset \mathbb{Z}_4$.
With the help of this homomorphism,
the standard inner product in $\mathbb{Z}_2^{\alpha}\times\mathbb{Z}_4^{\beta}$
is defined in \cite{BFPRV} as follows:
for any
$\mathbf{a}=(a_{0},\cdots, a_{\alpha-1}, a'_{0},\cdots, a'_{\beta-1})$
and
$\mathbf{b}=(b_{0},\cdots, b_{\alpha-1}, b'_{0},\cdots, b'_{\beta-1})$,
$$\mathbf{a}\cdot\mathbf{b}=2\left(\sum_{i=0}^{\alpha-1}a_i b_i\right)
 +\sum_{i=0}^{\beta-1}a'_i b'_i~\in~ \mathbb{Z}_{4}.$$
Let $C$ be any $\mathbb{Z}_2\mathbb{Z}_4$-additive code.
The {\em dual code} of $C$ is defined in the standard way by
$$C^{\bot}=\{\mathbf{a}\in\mathbb{Z}_2^{\alpha}\times\mathbb{Z}_4^{\beta}
  ~\mid~\mathbf{c}\cdot\mathbf{a}=0, ~\text{for all}~\mathbf{c}\in C\}.$$

Next, we introduce the definition of
$\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic codes
which is a natural extension of the definition of
classical cyclic codes.

\begin{definition}\label{acyclic}\rm\cite[Definition 3]{ASA}~
A subset $C$ of $\mathbb{Z}_2^{\alpha}\times\mathbb{Z}_4^{\beta}$
is called a {\em $\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic code} if
\begin{itemize}
\item[(i)] $C$ is an additive code;
\item[(ii)] For any codeword
${\bf c}=(c_0,c_{1},\cdots,c_{\alpha-1},c'_{0},c'_{1},\cdots,c'_{\beta-1})\in C$,
its cyclic shift
$\xi({\bf c})=(c_{\alpha-1},c_{0},\cdots,c_{\alpha-2},
  c'_{\beta-1},c'_{0},\cdots,c'_{\beta-2})$
is still in $C$.
\end{itemize}
\end{definition}

Note that in the definition the length $\alpha$ of the $\mathbb{Z}_2$-cycle
and the length $\beta$ of the $\mathbb{Z}_4$-cycle may be different.

Denote $R_{\alpha,\beta}=R_{\alpha}\times R'_{\beta}$,
where $R_{\alpha}=\mathbb{Z}_2[X]/\langle X^{\alpha}-1\rangle$
is the residue ring of the polynomial ring $\mathbb{Z}_2[X]$
modulo the ideal $\langle X^{\alpha}-1\rangle$ generated by
$X^{\alpha}-1$, and
$R'_{\beta}=\mathbb{Z}_4[X]/\langle X^{\beta}-1\rangle$.
Any element
${\bf a}=(a_{0},a_1,\cdots, a_{\alpha-1}, a'_{0},a'_1,\cdots, a'_{\beta-1})
 \in\mathbb{Z}_2^{\alpha}\times\mathbb{Z}_4^{\beta}$
can be identified with an element
${\bf a}(X)=(a(X), a'(X))\in R_{\alpha,\beta}$, where
$$\begin{array}{l}
a(X)= a_{0}+a_{1}X+\cdots+a_{\alpha-1}X^{\alpha-1}~\in~ R_\alpha,\\[3pt]
a'(X)= a'_{0}+ a'_{1}X+\cdots+a'_{\beta-1}X^{\beta-1}~\in~ R'_\beta.
\end{array}$$
This identification gives a one-to-one correspondence
between $\mathbb{Z}_2^{\alpha}\times\mathbb{Z}_4^{\beta}$
and $R_{\alpha,\beta}$.

For $f(X)\in \mathbb{Z}_4[X]$ and $(a(X), a'(X))\in R_{\alpha,\beta}$,
we define
$$f(X)\big(a(X), a'(X)\big)=\big(
\overline{f}(X)a(X),~ f(X)a'(X)\big)~\in~ R_{\alpha,\beta};$$
where $\overline{f}(X)$ is the reduction
$\overline{f}(X)= f(X)\pmod 2$, and
$\overline{f}(X)a(X)\in R_\alpha=\mathbb{Z}_2[X]/\langle X^{\alpha}-1\rangle$,
i.e., we should take the residue of $\overline{f}(X)a(X)$ mod $X^{\alpha}-1$;
the same is true of
$f(X)a'(X)\in R'_\beta=\mathbb{Z}_4[X]/\langle X^{\beta}-1\rangle$.
In a word, $R_{\alpha,\beta}$ is a $\mathbb{Z}_4[X]$-module.
It is shown in \cite{ASA} that
$\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic codes in
$\mathbb{Z}_2^{\alpha}\times\mathbb{Z}_4^{\beta}$
can be identified with $\mathbb{Z}_4[X]$-submodules of
$R_{\alpha,\beta}=\mathbb{Z}_2[X]/\langle X^{\alpha}-1\rangle
\times\mathbb{Z}_4[X]/\langle X^{\beta}-1\rangle$.

As is common in the discussion of usual cyclic codes, we write
codewords of a $\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic code $C$
as vectors or as polynomials interchangeably.
In either case, we use the same notation $C$ to denote the set of all codewords.
We follow this convention in the rest of the paper.

\section{A class of $\mathbb{Z}_2\mathbb{Z}_4-$additive cyclic codes}

From now on,
we always assume that $m$ is an odd positive integer.
For fundamentals on finite rings and coding theory,
please refer to \cite{HP, Mc}.

Following Section 2, we denote
$$R_m=\mathbb{Z}_2[X]/\langle X^{m}-1\rangle, \qquad
  R'_{m}=\mathbb{Z}_4[X]/\langle X^{m}-1\rangle, \qquad
  2\mathbb{Z}_4=\{0,2\}\subset\mathbb{Z}_4.
$$
We have seen that a subgroup $C$ of $\mathbb{Z}_2^m\times\mathbb{Z}_4^m$
is a $\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic code
if and only if $C$ is a $\mathbb{Z}_4[X]$-submodule of
$R_m\times R'_m$.
These codes are a special class
of $\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic codes.
We call such codes by $\mathbb{Z}_2\mathbb{Z}_4$-additive
cyclic codes with {\em cycle length $m$}.

Now we construct a class of $\mathbb{Z}_4[X]$-submodules of $R_m\times R'_m$.
Let
$$
2R'_{m}=\big\{a'(X)=\sum_{i=0}^{m-1}a'_iX^i\in R'_{m}~:~ 2\,|\,a'_i\big\},
$$
which is an ideal of $R'_m$, i.e.,
a $\mathbb{Z}_4[X]$-submodule of $R'_m$.
From the group homomorphism Eqn \eqref{2-map},
we have a $\mathbb{Z}_4[X]$-module isomorphism from $R_m$ onto $2R'_m$
as follows:
\begin{equation}\label{2R'}
 R_m \longrightarrow 2R'_m,~~
  \sum_{i=0}^{m-1}a_iX^i\longmapsto \sum_{i=0}^{m-1}2a_iX^i.
\end{equation}
Both $R_m$ and $2R'_m$ are in fact $\mathbb{Z}_2[X]$-modules
(and moreover, they are $R_m$-modules),
the above is a $\mathbb{Z}_2[X]$-isomorphism
(moreover, an $R_m$-isomorphism).

Thus, $R_m\times 2R'_{m}$ is a $\mathbb{Z}_4[X]$-submodule
of $R_m\times R'_{m}$. Hence any $\mathbb{Z}_2[X]$-submodule
of $R_m\times 2R'_{m}$,
i.e., any $R_m$-submodule of $R_m\times 2R'_{m}$,
is a $\mathbb{Z}_4[X]$-submodule of $R_m\times R'_{m}$.

Each element of $R_m\times 2R'_{m}$
is uniquely represented as
$$\big(a(X),2b(X)\big)\quad {\rm with}\quad
a(X)=\sum_{j=0}^{m-1}a_{j}X^{j},~
b(X)=\sum_{j=0}^{m-1}b_{j}X^{j}\in R_m.
$$
Any $R_m$-submodule of $R_m\times 2R'_{m}$
is generated by at most two elements. There are submodules of it
which cannot be generated by one element.
For example, $R_m\times 2R'_{m}$ as an $R_m$-submodule can not be generated
by one element, because: any $R_m$-module generated by one element
is of dimension over $\mathbb{Z}_2$ at most $m$.

For any given $\big(a(X),b(X)\big)\in R_m\times R_{m}$, let
\begin{equation}\label{C_a,b}
C_{a,b}=\big\{\big(f(X)a(X),2f(X)b(X)\big)\in R_m\times 2R'_{m}
 \,\big|\,f(X)\in R_m\big\},
\end{equation}
which is obviously an $R_m$-submodule of $R_m\times 2R'_{m}$,
hence is a $\mathbb{Z}_4[X]$-submodule of $R_m\times R'_{m}$;
in other words, $C_{a,b}$ is a $\mathbb{Z}_2\mathbb{Z}_4$-additive
cyclic code in $R_m\times R'_{m}$
generated by one element $\big(a(X),b(X)\big)$.
Since $C_{a,b}$ is an $R_m$-module,
it is a $\mathbb{Z}_2$-vector space,
and its dimension is denoted by $\dim C_{a,b}$.
We will exhibit good algebraic structures and
good asymptotic properties of the
$\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic codes
with cycle length $m$ defined above.

We show a way to construct a generator matrix
of the code $C_{a,b}$ in Eqn~\eqref{C_a,b}.
Recall that the element $\big(a(X),2b(X)\big)\in R_m\times 2R'_{m}$
is always identified with the word
$$(a_{0},a_{1},\cdots,a_{m-2},a_{m-1},~2b_{0},2b_{1},\cdots,2b_{m-2},2b_{m-1})
 \in \mathbb{Z}_2^m\times \mathbb{Z}_4^m. $$
We write
$$\begin{array}{l}
a(X)=a_0+a_1X+\cdots+ a_{m-1}X^{m-1},\\[5pt]
b(X)=b_0+b_1X+\cdots+b_{m-1}X^{m-1}.
\end{array}$$
For $a(X)$ we have an $m$-dimensional vector $(a_0,a_1,\cdots,a_{m-2},a_{m-1})$,
from which a circulant $m\times m$ matrix is constructed as follows:
$$
A=\begin{pmatrix}a_0 & a_1 & \cdots & a_{m-2} & a_{m-1} \\
 a_{m-1} & a_0 &\cdots &a_{m-3}& a_{m-2} \\
 \cdots & \cdots & \cdots & \cdots & \cdots\\
 a_{2} & a_{3} & \cdots & a_0& a_{1}\\
 a_{1}  & a_{2} & \cdots & a_{m-1} & a_0
\end{pmatrix}_{m\times m}.
$$
Similarly, from $b(X)$ we have a circulant $m\times m$ matrix as follows:
$$
B=\begin{pmatrix}b_0 & b_1 & \cdots & b_{m-2} & b_{m-1} \\
 b_{m-1} & b_0 &\cdots &b_{m-3}& b_{m-2} \\
 \cdots & \cdots & \cdots & \cdots & \cdots\\
 b_{2} & b_{3} & \cdots & b_0& b_{1}\\
 b_{1}  & b_{2} & \cdots & b_{m-1} & b_0
\end{pmatrix}_{m\times m}.
$$
Then we can construct an $m\times 2m$ matrix:
\begin{equation}\label{A A'}
 \widehat A=\left(\begin{array}{cc}
  A& 2B
 \end{array}\right)_{m\times 2m}.
\end{equation}
And it is easy to see that
$$
C_{a,b}=
\big\{(y_{0},y_1\cdots,y_{m-1})\widehat A\in
 \mathbb{Z}_2^{m}\times \mathbb{Z}_4^m
\,\big|\,(y_{0},y_1\cdots,y_{m-1})\in \mathbb{Z}_2^{m}\big\}.
$$
However, $\widehat A$ is not a generator matrix of $C_{a,b}$ in general,
since it is not of rank~$m$ in general.
To get a generator matrix from $\widehat A$,
we state a theorem as follows,
where $\gcd(\cdot,\cdot)$ denotes the greatest common divisor
and $\deg f(X)$ denotes the degree of the polynomial.

\begin{theorem}\label{thm2}
Given $(a(X),b(X))\in R_m\times R_{m}$ and $C_{a,b}$ as in Eqn \eqref{C_a,b}.
Let
\begin{equation}\label{g_a,a'}
 g_{a,b}(X)=\gcd\big(a(X),b(X),X^m-1\big)\quad \mbox{and}\quad
 h_{a,b}(X)=\frac{X^{m}-1}{g_{a,b}(X)}.
\end{equation}
Let $\big\langle g_{a,b}(X)\big\rangle_{R_m}$ be
the ideal of $R_m$ generated by $g_{a,b}(X)$.
Then $\dim C_{a,b}=\deg h_{a,b}(X)$, and
there is an $R_m$-module isomorphism
$\big\langle g_{a,b}(X)\big\rangle_{R_m}\cong C_{a,b}$
which maps $c(X)\in\big\langle g_{a,b}(X)\big\rangle_{R_m}$ to
$\big(c(X)a(X),\,2c(X)b(X)\big)\in C_{a,b}$.
\end{theorem}

{\bf Proof.}~  The following map is clearly an $R_{m}$-module homomorphism:
$$\gamma_{a,b}:~
 R_m\longrightarrow R_m\times 2R'_{m},\quad
 f(X)\longmapsto \big(f(X)a(X),\;2f(X)b(X)\big).
$$
Obviously, the image ${\rm im}(\gamma_{a,b})=C_{a,b}$.
Next we compute the kernel ${\rm ker}(\gamma_{a,b})$.
For $f(X)\in R_m$, $f(X)\in{\rm ker}(\gamma_{a,b})$
if and only if in $\mathbb{Z}_2[X]$ we have
\begin{equation}\label{system}
\left\{\begin{matrix}f(X)a(X)\equiv 0\pmod{X^{m}-1},\\
 f(X)b(X)\equiv 0\pmod{X^{m}-1}.\end{matrix}\right.
\end{equation}
The two equations are combined into one equation in $\mathbb{Z}_2[X]$:
$$f(X)\gcd\big(a(X),b(X)\big)\equiv 0~({\rm mod}~X^m-1),$$
from which we see that the system \eqref{system} holds if and only if
$$\textstyle
f(X)\equiv 0\quad \big({\rm mod}~\frac{X^{m}-1}{\gcd(a(X),b(X),X^m-1)}\big).
$$
So we get that
${\rm ker}(\gamma_{a,b})=\big\langle h_{a,b}(X)\big\rangle_{R_m}$.
Since $m$ is odd, $R_m$ is semisimple, i.e.,
$X^{m}-1$ has no repeated (multiple) roots in any extension of $\mathbb{Z}_2$.
Then we obtain a direct sum decomposition of $R_m$ as follows:
$$R_m=\big\langle h_{a,b}(X)\big\rangle_{R_m}\oplus
 \big\langle g_{a,b}(X)\big\rangle_{R_m}.
$$
Then the above $R_m$-homomorphism $\gamma_{a,b}$
induces an $R_m$-isomorphism:
$$\overline\gamma_{a,b}:~
 \big\langle g_{a,b}(X)\big\rangle_{R_m}
  \mathop{\longrightarrow}^{\cong} C_{a,b},\quad
 c(X)\longmapsto \big(c(X)a(X),\;2c(X)b(X)\big).
$$
In particular,
$$\dim C_{a,b}=\dim\big\langle g_{a,b}(X)\big\rangle_{R_m}
 =m-\deg g_{a,b}(X)=\deg h_{a,b}(X).\eqno\qed
$$

\begin{corollary}\label{cor3.2}
Let $C_{a,b}$, $h_{a,b}(X)$ be as in Theorem \ref{thm2} and
$r=\deg h_{a,b}(X)$, let $\widehat A$ be as in Eqn \eqref{A A'}.
Then the matrix consisting of the first $r$ rows of $\widehat A$
is a generator matrix of $C_{a,b}$.
\end{corollary}

{\bf Proof.}~  From the above theorem, we see that
the rank of $\widehat A$ is equal to $r$.
And by \cite[Theorem 3.6]{FL17}, the first $r$ rows of $\widehat A$
are linearly independent.
\qed

\begin{example}\label{exa1}\rm
 Take $m=3$, $a(X)=(X+1)^2=X^2+1$
and $b(X)=X+1$. By Theorem \ref{thm2},
$$\begin{array}{ll}
g_{a,b}(X)=\gcd\big(a(X),b(X),X^3-1\big)=X+1;\\[5pt]
 h_{a,b}(X)=(X^{3}-1)/g_{a,b}(X)=X^2+X+1.
\end{array}$$
So $\dim C_{a,b}=2$. Using the notations in Eqn \eqref{A A'}, we have
$$
 A=\begin{pmatrix}1&0&1\\ 1&1&0\\ 0&1&1 \end{pmatrix},
 \qquad B=\begin{pmatrix}1&1&0\\ 0&1&1\\1&0&1 \end{pmatrix};
$$
$$
\widehat A=\begin{pmatrix}1&0&1&2&2&0\\ 1&1&0&0&2&2\\ 0&1&1&2&0&2  \end{pmatrix}.
$$
Thus the first two rows of $\widehat A$ are linearly independent and
$$
 G=\begin{pmatrix} 1&0&1&2&2&0\\ 1&1&0&0&2&2\end{pmatrix}
$$
is a generator matrix of $C_{a,b}$. The $4$ codewords of $C_{a,b}$ are
as follows:
$$\begin{array}{cccc}
000000, & 110022,&101220, &011202.
\end{array}$$
\end{example}

\begin{example}\label{exa2}\rm
 Take $m=3$, $a(X)=X^2+X+1$
and $b(X)=X+1$. By Theorem \ref{thm2},
$$\begin{array}{ll}
g_{a,b}(X)=\gcd\big(a(X),b(X),X^3-1\big)
  =1;\\[5pt]
h_{a,b}(X)=(X^{3}-1)/g_{a,b}(X)=X^3-1.
\end{array}$$
So $\dim C_{a,b}=3$. And by the notations in Eqn \eqref{A A'},
$$
 A=\begin{pmatrix}1&1&1\\ 1&1&1\\ 1&1&1 \end{pmatrix},
 \qquad B=\begin{pmatrix}1&1&0\\ 0&1&1\\ 1&0&1\end{pmatrix};
$$
$$
\widehat A=\begin{pmatrix}1&1&1&2&2&0\\ 1&1&1&0&2&2\\ 1&1&1&2&0&2  \end{pmatrix}.
$$
Thus the rows of $\widehat A$ are linearly independent and
$$
 G=\left(\begin{array}{cccccc}
1&1&1&2&2&0\\ 1&1&1&0&2&2\\ 1&1&1&2&0&2
\end{array}\right)
$$
is a generator matrix of $C_{a,b}$. The $8$ codewords of $C_{a,b}$ are
as follows:
$$\begin{array}{cccccccc}
000000, & 111202, &111022, &000220,&111220,&000022, &000202, &111000.
\end{array}$$
\end{example}

Recall that $m$ is odd, for $X^m-1=(X-1)(X^{m-1}+\cdots+X+1)$, we get
\begin{equation}\label{a decomp Rm}
R_m=\mathbb{Z}_2[X]/\langle X^m-1\rangle
=\langle X^{m-1}+\cdots+X+1\rangle_{R_m}\oplus \langle X-1 \rangle_{R_m},
\end{equation}
where $\langle X^{m-1}+\cdots+X+1\rangle_{R_m}$ is the trivial code
of dimension $1$. Thus, in the following we
consider the following ideal of $R_m$ generated by $X-1$:
\begin{equation}\label{J}
  J_m=\langle X-1 \rangle_{R_m},
\end{equation}
which is called the {\em augmentation ideal} of $R_{m}$ in
the representation theory of finite groups.
Similarly, in $R'_m=\mathbb{Z}_4[X]/\langle X^m-1\rangle$
we denote $J'_m=\langle X-1\rangle_{R'_m}$.
It is easy to see that
$2J'_m=\{2f(X)\mid f(X)\in J'_m\}=\langle 2(X-1)\rangle_{R'_m}$,
and the $R_m$-isomorphism in Eqn \eqref{2R'}
induces an $R_m$-module isomorphism:
\begin{equation}\label{JJ'}
 J_m\times J_m \mathop{\longrightarrow}^{\cong} J_m\times 2J'_m,\quad
   (a(X),b(X))\longmapsto (a(X),2b(X)).
\end{equation}

For the code $C_{a,b}$ constructed in Eqn~\eqref{C_a,b}
where $a(X)$, $b(X)$ chosen from $R_m$,
in the following we will consider
only the $a(X)$, $b(X)$ chosen from $J_{m}$.
In that case when $a(X), b(X)\in J_m$, by Theorem \ref{thm2},
the $\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic code $C_{a,b}$
with cycle length $m$
can be reformulated within $J_{m}$ as follows:
\begin{equation}\label{JC_a,b}
C_{a,b}=\big\{(c(X)a(X),2c(X)b(X))\in R_m\times 2R'_{m}
 \;\big|\;c(X)\in J_{m}\big\}\subseteq J_m\times 2J'_m.
\end{equation}

The class of the $\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic code
 $C_{a,b}$ (of cycle length $m$) for $a(X),b(X)\in J_m$
 will play a key role in the next section.

\begin{lemma}\label{dC_a,b}
Let notation be as above,
let $\big(a(X),b(X)\big)\in J_{m}\times J_{m}$. Then
$\dim C_{a,b}\le m-1$; and $\dim C_{a,b}<m-1$
if and only if there is an irreducible factor
$p(X)$ of $X^{m-1}+\cdots+X+1$ in $\mathbb{Z}_2[X]$
such that $p(X)\,|\,a(X)$ and $p(X)\,|\,b(X)$.
\end{lemma}

{\bf Proof.}~  By the definition of $J_{m}$ in Eqn~\eqref{J} we have
$$(X-1)\,\big|\;g_{a,b}(X)=\gcd\big(a(X),b(X),X^m-1\big);$$
where $g_{a,b}(X)$ is defined in Eqn \eqref{g_a,a'}. So
$$
\big\langle g_{a,b}(X)\big\rangle_{R_m}\subseteq
\big\langle X-1\big\rangle_{R_m}=J_{m}.
$$
By Theorem \ref{thm2},
$\dim C_{a,b}=\dim\big\langle g_{a,b}(X)\big\rangle_{R_m}\le\dim J_m=m-1$.
And,
$\dim C_{a,b}=m-1$ if and only if
$$g_{a,b}(X)=\gcd\big(a(X),b(X),X^m-1\big)=X-1,$$
if and only if there is no irreducible factor
$p(X)$ of $\frac{X^m-1}{X-1}$
such that $p(X)\,|\,a(X)$ and $p(X)\,|\,b(X)$.
\qed

\begin{lemma}\label{number I}
Let $\ell_m$ be the minimal degree of irreducible factors
in $\mathbb{Z}_2[X]$ of $\frac{X^m-1}{X-1}$,
and $d$ be an integer with $\ell_m\le d<m$.
Then any non-zero ideal of $R_m$ contained in $J_{m}$
is of dimension at least $\ell_m$, and the number of the ideals
of dimension $d$ which are contained in $J_{m}$
is at most $m^{\frac{d}{\ell_m}}$.
\end{lemma}

{\bf Proof.}~  By the decomposition in Eqn \eqref{a decomp Rm}
and the definition of $J_m$ in Eqn \eqref{J},
each irreducible ideal contained in $J_{m}$ is corresponding to exactly
one irreducible divisor of $\frac{X^m-1}{X-1}$ such that the dimension
of the ideal is equal to the degree of the corresponding divisor. Thus,
the minimal dimension of the ideals contained in $J_{m}$ is equal to $\ell_m$.
And, any $d$-dimensional ideal contained in $J_{m}$
is a sum of at most $d/\ell_m$ irreducible ideals.
So the number of the $d$-dimensional ideals contained in $J_{m}$
is at most the partial sum of binomial coefficients
$\sum_{i=1}^{\lfloor d/\ell_m\rfloor}{h\choose i}$,
where $h$ is the number of the irreducible ideals contained in $J_{m}$
 and $\lfloor d/\ell_m\rfloor$ denotes
the largest integer which is not larger than $d/\ell_m$.
It is easy to check that
\begin{eqnarray*}
\sum_{i=1}^{\lfloor d/\ell_m\rfloor}{h\choose i}&\leq& \sum_{i=1}^{\lfloor d/\ell_m\rfloor}{m\choose i}\\
&\leq&\frac{d}{\ell_m}\cdot\frac{m(m-1)\cdots(m-\frac{d}{\ell_m}+1)}{1\cdot2\cdots\frac{d}{\ell_m}}\\
&\leq&m^{\frac{d}{~\ell_m}}~.
\end{eqnarray*}
\qed

\section{Random $\mathbb{Z}_2\mathbb{Z}_4-$additive cyclic codes}

Keep the notations in Section 2.

For a real number $x$ with $0\le x\le 1$, let
 $H(x)=-x\log_2 x-(1-x)\log_2(1-x)$
with the convention that $0\log_q0=0$.
The function $H(x)$ is called the (binary) {\em entropy}.
Note that $H(x)$ is a strictly increasing concave function
in the interval $[0,1/2]$ with $H(0)=0$ and $H(1/2)=1$,
see \cite[\S 2.10.6]{HP}.
In the following, we assume that $\delta$ is a real number such that
\begin{equation}\label{delta}
0<\delta< 1/3 \qquad\mbox{and}\qquad H(3\delta/2)< 1/2.
\end{equation}
Note that, if $x\in[0,1/2]$ such that $H(x)=1/2$, then $0.11<x<0.1101$.
So the assumption (4.1) is about to say that
$0<\delta\le 0.0733$.

As in Section 3, we denote
$$ R_m=\mathbb{Z}_2[X]/\langle X^m-1\rangle,\qquad
 R'_m=\mathbb{Z}_4[X]/\langle X^m-1\rangle,
$$
and (see Eqn \eqref{J} and Eqn \eqref{JJ'})
$$J_m=\langle X-1\rangle_{R_m}, \qquad
2J'_m=\langle 2(X-1)\rangle_{R'_m}=\{2b(X)\mid b(X)\in J_m\}.$$
In this section we view the set $J_{m}\times 2J'_{m}$
as a probability space, whose samples are afforded with equal probability.
For each sample $\big(a(X),2b(X)\big)\in J_{m}\times 2J'_{m}$
(with $b(X)\in J_m$), by Eqn \eqref{JC_a,b}
we have a $\mathbb{Z}_2\mathbb{Z}_4-$additive cyclic code
with cycle length $m$:
\begin{equation}\label{JJC_a,b}
C_{a,b}=\big\{(c(X)a(X),2c(X)b(X))\in R_m\times 2R'_{m}
 \,\big|\,c(X)\in J_{m}\big\}.
\end{equation}
Then $C_{a,b}$ is a random code over
the probability space $J_{m}\times 2J'_{m}$,
hence the relative minimum distance $\Delta(C_{a,b})$ of $C_{a,b}$
 (see Eqn \eqref{def relat dis}) and the dimension $\dim(C_{a,b})$ of $C_{a,b}$
are random variables over the probability space.
We are interested in the asymptotic properties
of the probabilities ${\Pr\big(\Delta(C_{a,b})>\delta\big)}$
and ${\Pr\big(\dim C_{a,b}=m-1\big)}$.
As a preparation, we introduce a type of $0$-$1$-variables.

Recall that a random variable  is called a {\em Bernoulli variable}
if it takes the value $1$ with probability $p$
and the value $0$ with probability $1-p$.
Given any $c(X)\in J_{m}$. We define a Bernoulli variable $Y_c$
over the probability space $J_{m}\times 2J'_{m}$ as follows:
for samples $\big(a(X),2b(X)\big)\in J_{m}\times 2J'_{m}$,
\begin{equation}\label{Y_c}
Y_c=\begin{cases} 1, & 1\le{\rm wt}\big(c(X)a(X),2c(X)b(X)\big)\le 3m\delta;\\
 0, & {\rm otherwise}.\end{cases} \quad
\end{equation}
Since $c(X)\in J_{m}$, the set
$\{c(X)a(X)\in R_m\,|\,a(X)\in J_{m}\}$ is the ideal of $R_m$
generated by $c(X)$, we denote it by $I_c$ and denote its dimension by $d_c$,
i.e.,
\begin{equation}\label{I_c}
 I_c=\big\langle c(X)\big\rangle_{R_m}\subseteq J_{m}, \quad d_c=\dim I_c\,.
\end{equation}

By $|S|$ we denote the cardinality of any set $S$.

\begin{lemma}\label{IcIc}
Let notation be as in Eqns \eqref{delta}--\eqref{I_c}.
For $I_c\times I_c\subseteq R_m\times R_m$, let
$$(I_c\times I_c)^{\le 3m\delta}=
  \big\{\big(f(X),g(X)\big)\in I_c\times I_c\,\big|\;
  {\rm wt}_H\big(f(X),g(X)\big)\le 3m\delta \big\}.$$
Then
$$ \big|(I_c\times I_c)^{\le 3m\delta}\big|\le 2^{2d_c H(3\delta/2)}. $$
\end{lemma}

{\bf Proof.}~  Note that the length of the code
$I_c\times I_c$ is $2m$, i.e., $\dim(R_m\times R_m)=2m$.
And the dimension $\dim(I_c\times I_c)=2d_c$.
The fraction of $3m\delta$ over the length is
$(3m\delta)/(2m)=3\delta/2$, and by the assumption Eqn \eqref{delta},
$0<3\delta/2<1/2$. So
the inequality follows immediately from
\cite[Remark~3.2 and Corollary~3.5]{FL}.
\qed

With the notation as in Eqns \eqref{delta}--\eqref{I_c},
we show an estimation of the expectation ${\rm E}(Y_c)$
of the random variable $Y_c$ for later quotation.

\begin{lemma}\label{EY_c}~
${\rm E}(Y_c)\le 2^{-2d_c+2d_c H(3\delta/2)}$.
\end{lemma}

{\bf Proof.}~
In $2J'_{m}\subset R'_m$, we have an ideal
$$
 I'_c=\big\langle 2c(X)\big\rangle_{R'_{m}}
 =\{2c(X)b(X)\in R'_{m}\,|\,b(X)\in J_{m}\} \subseteq 2J'_{m}.
$$
By Eqn \eqref{JJ'}, we see that:
\begin{equation}\label{Ic=I'c}
I'_c\cong I_c, \qquad\mbox{and}\qquad
 \dim I'_c=\dim I_c=d_c\,.
\end{equation}
For $I_c\times I'_c\subseteq R_m\times R'_m$, we denote
$$(I_c\times I'_c)^{\le 3m\delta}=
  \big\{\big(f(X),2g(X)\big)\in I_c\times I'_c\,\big|\;
  {\rm wt}\big(f(X),2g(X)\big)\le 3m\delta \big\}.$$
Since $Y_c$ is a $0$-$1$-variable, the expectation of $Y_c$
is just the probability that $Y_c=1$, So
\begin{equation}\label{EYc}
 {\rm E}(Y_c)=\Pr(Y_c=1)=
 \frac{\big|(I_c\times I'_c)^{\le 3m\delta}\big|-1}
 {|I_c\times I'_c|}.
\end{equation}
For $f(X),g(X)\in R_m$, we have seen that
$${\rm wt}_L(2g(X))=2\,{\rm wt}_H(g(X)),$$
where the weight in the left hand side is in $R'_m$,
while the weight in the right hand side is in $R_m$; so
\begin{equation*}
{\rm wt}(f(X),2g(X))\ge {\rm wt}_H(f(X),g(X)).
\end{equation*}
Thus, with notation in Lemma \ref{IcIc}, we get
\begin{equation}\label{ww}
\big|(I_c\times I'_c)^{\le 3m\delta}\big|
\le \big|(I_c\times I_c)^{\le 3m\delta}\big|.
\end{equation}
By Eqn \eqref{EYc}, Eqn \eqref{ww}, Eqn \eqref{Ic=I'c} and Lemma \ref{IcIc},
$$
 {\rm E}(Y_c)\le
 \frac{\big|(I_c\times I_c)^{\le 3m\delta}\big|}
 {|I_c\times I_c|}\le \frac{2^{2d_cH(3\delta/2)}}{2^{2d_c}}
 =2^{-2d_c+2d_c H(3\delta/2)}.
 \eqno\qed
$$

Now we can present an estimation of the probability
that the relative minimum distance $\Delta(C_{a,b})$ is at most $\delta$.

\begin{lemma}\label{EX}
Let $\delta$ be as in Eqn \eqref{delta} and $C_{a,b}$ be as in
Eqn \eqref{JJC_a,b}. Let $\ell_m$ be the minimal degree of
the irreducible divisors of $\frac{X^m-1}{X-1}$ as in Lemma \ref{number I}.
Then
$$ \Pr\big(\Delta(C_{a,b})\le\delta\big)\le
\sum_{j=\ell_m}^{m-1}
 2^{-2j\big(\frac{1}{2}-H(\frac{3\delta}{2})-\frac{\log_2 m}{\ell_m}\big)}.
$$
\end{lemma}

{\bf Proof.}~
Let $Y_c$ for $c(X)\in J_{m}$ be the $0$-$1$-variable
defined in Eqn \eqref{Y_c}. Let
$$Y=\sum_{c(X)\in J_{m}} Y_c\,.$$
Then $Y$ is a non-negative integer random variable
over the probability space $J_{m}\times 2J'_{m}$.
By Eqn \eqref{JJC_a,b} and Eqn \eqref{Y_c}, $Y$ stands for
the number of $c(X)\in J_{m}$ such that the codeword
$\big(c(X)a(X), 2c(X)b(X)\big)$ of $C_{a,b}$ is non-zero with weight
at most $3m\delta$. By Eqn \eqref{def relat dis},
$\Delta(C_{a,b})={\rm wt}(C_{a,b})/3m$. We get
$$
 \Pr\big(\Delta(C_{a,b})\le\delta\big)=\Pr\big(Y>0\big).
$$
By a Markov's inequality \cite[Theorem 3.1]{MU},
$\Pr\big(Y>0\big)\le {\rm E}(Y)$. So we can prove the lemma by
estimating the expectation ${\rm E}(Y)$.

By the linearity of the expectation,
${\rm E}(Y)=\sum_{c(X)\in J_{m}}{\rm E}(Y_c)$.
For any ideal $I$ of $J_{m}$ (we denote it by $I\le J_{m}$), let
$I^*=\big\{c(X)\in I\,\big|\,I_c=I\big\}$, where
$I_c$ is defined in Eqn \eqref{I_c}. That is,
$$
I^*=\big\{c(X)\in I\,\big|\,d_c=\dim I\big\},
$$
where $d_c=\dim I_c$, see Eqn \eqref{I_c}.
It is easy to see that
$ J_{m}=\bigcup_{I\le J_{m}}I^*$, where the subscript
``$I\le J_{m}$'' means that $I$ runs over the ideals contained in $J_{m}$.
By Lemma \ref{number I}, if $0\ne I\le J_{m}$
then $\ell_m\le\dim I\le m-1$. So
$$
{\rm E}(Y)=\sum_{I\le J_{m}}\sum_{c(X)\in I^*}{\rm E}(Y_c)
=\sum_{j=\ell_m}^{m-1}
  \sum_{\mbox{\tiny$\begin{array}{c}I\le J_{m}\\ \dim I=j\end{array}$}}
   \sum_{c(X)\in I^*}{\rm E}(Y_c).
$$
For $I\le J_{m}$ with $\dim I=j$, by Lemma \ref{EY_c}
and the fact that $|I^*|\le |I|=2^j$, we get
$$
\sum_{c(X)\in I^*}{\rm E}(Y_c)\le \sum_{c(X)\in I^*}
  2^{-2j+2j H(3\delta/2)} \le 2^{-j+2j H(3\delta/2)}.
$$
By Lemma \ref{number I} again, the number of
$I\le J_{m}$ with $\dim I=j$ is less that $m^{j/\ell_m}$.
And note that $\log_2 m\le\frac{j\log_2 m}{\ell_m}$ (as $j\ge\ell_m)$.
So
\begin{eqnarray*}
{\rm E}(Y)&\le& \sum_{j=\ell_m}^{m-1} m^{j/\ell_m} 2^{-j+2j H(3\delta/2)}\\
&=& \sum_{j=\ell_m}^{m-1}
 2^{-2j\big(\frac{1}{2}- H(\frac{3\delta}{2})-\frac{\log_2 m}{\ell_m}\big)}.
\end{eqnarray*}
The lemma is proved.
\qed

\medskip
By \cite[Lemma 2.6]{BM}, there are positive integers $m_1$, $m_2$, $\cdots$
satisfying that
\begin{equation}\label{m_i}
m_i ~\text{is odd }, \quad m_i\to\infty,\quad\lim\limits_{i\to\infty}\frac{\log_2 m_i}{\ell_{m_i}} =0,
\end{equation}
where $\ell_{m_i}$ is the minimal degree of
the irreducible divisors of $\frac{X^{m_i}-1}{X-1}$
as defined in Lemma \ref{number I}.
For each $m_i$, let
\begin{equation}\label{JJC_a,bi}
C^{(i)}_{a,b}=\big\{(c(X)a(X),2c(X)b(X))\in R_{m_i}\times 2R'_{m_i}
 \,\big|\,c(X)\in J_{m_i}\big\}
\end{equation}
be the random $\mathbb{Z}_2\mathbb{Z}_4-$additive cyclic code
with cycle length $m_i$ as defined in Eqn~\eqref{JJC_a,b}.

\begin{theorem}\label{Delta C}
Let notation be as above. Assume that $0<\delta<1/3$ and
$H(3\delta/2)<1/2$. Then
$$
 \lim\limits_{i\to\infty}\Pr\big(\Delta(C^{(i)}_{a,b})>\delta\big)=1.
$$
\end{theorem}

{\bf Proof.}~
By the assumption (i.e. the assumption in Eqn \eqref{delta}), we have
$\frac{1}{2}-H(\frac{3\delta}{2})>0$.
By Eqn~\eqref{m_i}, there are a positive real number $\beta$
and an integer $N$ such that
$$\textstyle
 \frac{1}{2}-H(\frac{3\delta}{2})-\frac{\log_2 m_i}{\ell_{m_i}}
 \ge \beta,\qquad \forall~ i>N.
$$
By Lemma \ref{EX},
\begin{eqnarray*}
\lim\limits_{i\to\infty}\Pr\big(\Delta(C^{(i)}_{a,b})\le\delta\big)
\kern-3pt&\le&\kern-3pt\lim\limits_{i\to\infty}
\sum_{j=\ell_{m_i}}^{m_i-1}2^{-2j\beta}
\le \lim\limits_{i\to\infty}
\sum_{j=\ell_{m_i}}^{m_i-1}2^{-2\ell_{m_i}\beta}\\
&\le&\kern-3pt\lim\limits_{i\to\infty}m_i 2^{-2\ell_{m_i}\beta}
=\lim\limits_{i\to\infty}
2^{-2\ell_{m_i}\big(\beta-\frac{\log_2 m_i}{2\ell_{m_i}}\big)}.
\end{eqnarray*}
Since $\lim\limits_{i\to\infty}\frac{\log_2 m_i}{2\ell_{m_i}}=0$
(which implies that $\lim\limits_{i\to\infty}\ell_{m_i}=\infty$),
we obtain that
$\lim\limits_{i\to\infty}\Pr\big(\Delta(C^{(i)}_{a,b})\le \delta)=0$.
\qed

Next, we estimate the rate
$R(C^{(i)}_{a,b})$ of the random code $C^{(i)}_{a,b}$.

\begin{theorem}\label{dim C}
Let $m_1$, $m_2$, $\cdots$ be positive integers satisfying Eqn \eqref{m_i},
and let $C^{(i)}_{a,b}$ be as in Eqn \eqref{JJC_a,bi}. Then
$$
 \lim\limits_{i\to\infty}\Pr\big(\dim C^{(i)}_{a,b}=m_i-1\big)=1.
$$
\end{theorem}

{\bf Proof.}~  For any given $i$,
let $X^{m_i-1}+\cdots+X+1=p_1(X)\cdots p_{h_i}(X)$
be the irreducible decomposition in $\mathbb{Z}_2[X]$,
i.e., $p_1(X),\cdots p_{h_i}(X)$ are distinct irreducible polynomials
in $\mathbb{Z}_2[X]$. By Chinese Remainder Theorem,
there is a natural isomorphism:
\begin{equation}\label{J iso}
R_{m_i}\cong
\mathbb{Z}_2[X]/\langle X-1\rangle\times
 \mathbb{Z}_2[X]/\langle p_1(X)\rangle\times\cdots
  \times \mathbb{Z}_2[X]/\langle p_{h_i}(X)\rangle.
\end{equation}
We restrict it to the augmentation ideal $J_{m_i}\subset R_{m_i}$
and get a natural isomorphism:
$$
J_{m_i}=\langle X-1\rangle_{R_{m_i}}\cong
 \mathbb{Z}_2[X]/\langle p_1(X)\rangle\times\cdots
  \times \mathbb{Z}_2[X]/\langle p_{h_i}(X)\rangle
$$
which maps $a(X)\in J_{m_i}$ to
$$
\big(\mu_{m_i}^{(1)}(a(X)),\cdots,\mu_{m_i}^{(h_i)}(a(X))\big)\in
\mathbb{Z}_2[X]/\langle p_1(X)\rangle\times\cdots
  \times \mathbb{Z}_2[X]/\langle p_{h_i}(X)\rangle,
$$
where
$$
 \mu_{m_i}^{(j)}(a(X))=a(X)~({\rm mod}~{p_j(X)})~~ \in~
  \mathbb{Z}_2[X]/\langle p_j(X)\rangle, \qquad j=1,\cdots, h_i.
$$
Let $a(X),b(X)\in J_{m_i}$. By Lemma \ref{dC_a,b},
$\dim C^{(i)}_{a,b}\le m_i-1$, and $\dim C^{(i)}_{a,b}<m_i-1$
if and only if there is an index $j$ with $1\le j\le h_i$ such that
$\mu_{m_i}^{(j)}(a(X))=0=\mu_{m_i}^{(j)}(b(X))$.
In other words, $\dim C^{(i)}_{a,b}=m_i-1$
if and only if for any $j=1,\cdots,h_i$, in
$\mathbb{Z}_2[X]/\langle p_j(X)\rangle
  \times\mathbb{Z}_2[X]/\langle p_j(X)\rangle$
the following holds:
\begin{equation}\label{mu,mu}
\big(\mu_{m_i}^{(j)}(a(X)),\,\mu_{m_i}^{(j)}(b(X))\big)\ne (0,0).
\end{equation}
Let $d_j=\deg p_j(X)$. Then $\mathbb{Z}_2[X]/\langle p_j(X)\rangle$
is a finite field of cardinality $2^{d_j}$.
Note that
$\mu_{m_i}^{(j)}: J_{m_i}\to \mathbb{Z}_2[X]/\langle p_j(X)\rangle$
is surjective homomorphism, cf. the isomorphism in Eqn \eqref{J iso}.
So the probability that Eqn~\eqref{mu,mu} holds is equal to
$\frac{2^{2d_j}-1}{2^{2d_j}}=1-2^{-2d_j}$.
Obviously,  the events that Eqn~\eqref{mu,mu}
holds for $j=1,\cdots,h_i$ are randomly independent. Thus
$$
\Pr\big(\dim C^{(i)}_{a,b}=m_i-1\big)=\prod_{j=1}^{h_i}(1-2^{-2d_j}).
$$
By definition of $\ell_{m_i}$ in Eqn \eqref{m_i},
$\ell_{m_i}\le d_j$ for $j=1,\cdots,h_i$,
hence $h_i\le \frac{m_i-1}{\ell_{m_i}}\le \frac{m_i}{\ell_{m_i}}$.
Thus $\Pr\big(\dim C^{(i)}_{a,b}=m_i-1\big)\ge
\big(1-2^{-2\ell_{m_i}}\big)^{\frac{m_i}{\ell_{m_i}}}$, i.e.,
\begin{equation*}
\Pr\big(\dim C^{(i)}_{a,b}=m_i-1\big)~\ge~
 \big(1-2^{-2\ell_{m_i}}\big)^
  {2^{2\ell_{m_i}}\cdot\frac{m_i}{\ell_{m_i}2^{2\ell_{m_i}}}}.
\end{equation*}

Now we consider the asymptotic case when $i\to\infty$.
By the assumption in Eqn \eqref{m_i},
$\lim\limits_{i\to\infty}\frac{\log_2 m_i}{\ell_{m_i}}=0$
(which implies that $\lim\limits_{i\to\infty}\ell_{m_i}=\infty$).
So
$$\lim\limits_{i\to\infty}\frac{m_i}{\ell_{m_i}2^{2\ell_{m_i}}}
 =\lim\limits_{i\to\infty} 2^{-\ell_{m_i}\big(2-
  \frac{\log_2 m_i}{\ell_{m_i}}+\frac{\log_2\ell_{m_i}}{\ell_{m_i}}\big)}
 =0.
$$
Note that $\big(1-2^{-2\ell_{m_i}}\big)^{2^{2\ell_{m_i}}}>1/4$.
We get that
$$
\lim\limits_{i\to\infty}\Pr\big(\dim C^{(i)}_{a,b}=m_i-1\big)\ge
\lim\limits_{i\to\infty}(1/4)^{\frac{m_i}{\ell_{m_i}2^{2\ell_{m_i}}}}=1.
\eqno\qed
$$

\begin{theorem}\label{a g}
Let $\delta$ be a real number
such that $0<\delta< 1/3$ and $H(3\delta/2)<\frac{1}{2}$.
Then there is a sequence of $\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic
codes $C_i$ with cycle length $m_i$ for $i=1,2,\cdots$
such that $m_i\to\infty$ and the following hold.

(i)~ $\lim\limits_{i\to\infty}R(C_i)=\frac{1}{3}$;

(ii)~ $\Delta(C_i)>\delta$ for all $i=1,2,\cdots$.
\end{theorem}

{\bf Proof.}~  Note that $R(C_i)=\frac{\dim C_i}{3m_i}$, see Eqn \eqref{def rate}.
From Theorem~\ref{Delta C} and Theorem~\ref{dim C},
there is an integer $N\ge 0$ such that for any $i>N$ we can get a
$\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic code $C_i$ with cycle length $m_i$
such that $\Delta(C_i)>\delta$ and $R(C_i)=\frac{1}{3}-\frac{1}{3m_i}$.
Removing the first $N$ codes and relabeling the remaining codes,
we obtain the conclusions at once.
\qed

\begin{remark}\rm
For example, if we take $H(3\delta/2)=0.4999$, then $\delta\approx 0.0733$,
and we get a sequence $C_1,C_2,\cdots$ of $\mathbb{Z}_2\mathbb{Z}_4$-additive
cyclic codes with cycle length going to infinity such that
$R(C_i)\to 1/3$, and $\Delta(C_i)>0.0733$ for all $i=1,2,\cdots$.
\end{remark}

\begin{remark}\rm
The probabilistic techniques used in this paper
might be applied to other families of codes,
provided one could find a class of codes in the family
which possess suitable properties
such that the probabilistic method can work well.
\end{remark}

\section*{Acknowledgements}
The research of the authors is supported by NSFC
with grant numbers 11271005.
The authors are grateful to the reviewers for their
careful and suggestive comments which are
helpful in creating the improved final version.

\small

redpajama_set_name: RedPajamaArXiv


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My wife and daughter are pretty smart and I am seldom surprised with some of the things that they will come people, up with.
"Beware of the Ides of March" smirked my daughter.
"The ides are the middle of March," answered my daughter. "There is an ides of January and February as well, in fact, there is an ides of every month."
"So I am supposed to beware of the middle of the month?"
"Not you, Dad; Julius Caesar was supposed to be careful. That was the day, Shakespeare and history tell us that Brutus stabbed him."
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"March is a good month Dad, and the middle of March is a good time for most of us."
"'Cept for Caesar," I replied.
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"Why would the Romans name March after their God of war?"
"You've studied history Dad; Spring was an ideal time for armies to travel. Most of the bad winter weather is over and snow, particularly in mountain passes, normally starts to melt. It was also a time when food could be more easily foraged.
"I will remember that, next time I decide to start a war."
I picked up an Almanac. "I am sure that there are a lot of important dates in March. let's see: March 1, 1872-Yellowstone National Park; March 2, 1904- Dr. Seuss born; March 3, 1847-Alexander Graham Bell; March 4, 1789-First meeting of Congress; March 5, 1770-Boston Massacre; March 6, 1912-Oreo Cookies; March 7, 1849-Luther Burbank; March 8, 1775-Thomas Paine calls for emancipation of all slaves; March 9, 1454-Amerigo Vespucci born; March 10, 1862-first paper money issued; March 11, Johnny Appleseed Day…"
My wife walked into our living room and asked what we were doing.
I looked up and said "Just looking though an old Almanac…"
My daughter chirped: "We were discussing the Ides of March, but Dad hasn't made it to the Ides yet."
"I am working on it. Have you looked at some of the designations that they have for march?
The first week of March is Returning your Borrowed books week, and the second week is National Procrastinators week, followed by clean your closet week."
My daughter grinned "Dad, you made that up."
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My wife took the Almanac away from both of us and looked with amazement at some of the designations. "Youth Art Month, National Peanut Month, Women's History Month, Music in our Schools Month, March madness…the list seems to go on and on. Who in the world makes up these National Month Designations?"
I grinned. "Different groups; last month a Pizza chain declared February National Football Month. Sometimes different media publicans will make a declaration, other times Congress, or even the President will make a special declaration. They will proclaim a day, a week or a month special. It can be a one-time thing or an ongoing event. It can be done to honor an idea, an act, a person, or just for fun. Hey, a lot of things happened in March. The rubber band was invented, the pencil eraser was patented, the first map of the US was published, Uranus was discovered, The Barbi doll invented and every March the Swallows return to San Juan Capistrano.
My daughter looked at me "I am not sure that the whole world knows about the Swallows returning to Capistrano."
"That may be so, but National Geographic once did a story about them."
My Irish wife brightened. "St. Patrick's day is celebrated in March."
I interjected: "So is St. Urho's day. St. Urho is the patron saint of Finland."
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"Uuh, Finnish willingness to get right in there and do things?"
My wife glared at me: "Like What?
"Ravenous grasshoppers besieged the Finnish grape crop. St. Urho fearlessly waved his pitch fork and said: 'Heinäsirkka, heinäsirkka, mene täältä hiiteen', (which means something like Grasshoppers grasshoppers get the hell out of here) and all the grasshoppers fled to Russia, Latvia and elsewhere. Poisonous frogs plagued Finland. Urho lifted his pitchfork, drove them into a lake and let the lake freeze. Then he instructed the Finns to cut the ice out of the lake and ship it all to France. From that time on, French have always liked frogs legs.
There are statues of Urho all over the world. St. Urho's day is always one day before St. Patrick's. This is by design. By having his birthday one day early, the Fins can get to the beer well before the Irish."
My daughter grinned "Urho's day is pretty close to the Ides of March. I think you had better beware…"
"Okay, then let us all beware of the Ides.


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text:
"Intimités" Lecture de poèmes par Ashok Vajpeyi et Thierry Pillon
Mardi 29 juin 2010 à 18h IEA de Nantes, Amphithéâtre Simone Weil, 5 allée Jacques Berque à Nantes
Ashok Vajpeyi, a Hindi poet-critic, translator, editor and culture-activist, is a major cultural figure of India. With more thant 13 books of poetry, 7 of criticism in Hindi and 4 bookds on art and music in English to his credit, he is widely recognized as an outstanding promoter of culture and an innovative institution-builder. Over the years he has worked tirelessly to enhance the mutual awareness and interaction between Indian and foreign cultures. A frequent presence at some of the major conferences, seminars and poetry-festivals, he has raised his voice for the autonomy of literature and arts as against contemporary tyrannies of ideologies, markets and fundamentalism. As editor of many prestigious journals he has done much to promote critical awareness of contemporary and classical arts and young talent in poetry and criticism. As an organizer he has more than a thousands events to his credit relating to literature, music, dances, theatre, visual arts, fold and tribal arts, cinema etc. He has been awarded the Sahitya Akademi Award, the Dayawati Kavi Shekhar Samman and the Kabir Samman.
He has written two large books on the Paris-based Indian master Sayed Haider Raza and also one on 7 contemporary Indian abstract painters. He set up the renowned multi-arts center Bharat Bhavan in Bhopal; has been the first Vice-Chancellor of Mahatma Gandhi International Hindi University (set up by Govt. of India). For more than a year he doubled up as the Director General of the National Museum, New Delhi and as the Vice-Chairman National Museum of Man, Bhopal.
A prominent public intellectual of India, he has been a creative global-trotter and visited Europe many times to attend conferences, deliver lectures and give readings. Has been a writer in-residence at Jamia Millia Islamia University and a fellow of KK Birla Foundation. He lives in Delhi after retiring from civil service. Ha has been decorated by the President of Republic of Poland b the outstanding national award "The Officer's Cross of Merit of the Republic of Poland", the French Govt. by the award of "Officier de l'Ordre des Arts et des Des Letters" and the Embassy of Romania to India by the award "Best Promoter of Indo-Romanian Relations".
Presently Chairman, Lalit Kala Akademi, Ministy of Culture, Govt. of India, New-Delhi.


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text:
Donald Cameron (ca. 1836 – in or after 1882) was a farmer, official and political figure on Prince Edward Island. He represented 1st Queens in the Legislative Assembly of Prince Edward Island from 1867 to 1873 and from 1879 to 1882 as a Conservative.

He was born in Springton, Prince Edward Island, the son of John Cameron, a Scottish immigrant, and Mary Stuart. He was educated there and settled in Springton. Cameron was defeated when he ran for reelection in 1873 and again in 1882. He was postmaster for Township 67 and a commissioner for taking affidavits in Supreme Court.

References

1830s births
1880s deaths
Year of death missing
Year of birth uncertain
Progressive Conservative Party of Prince Edward Island MLAs
People from Queens County, Prince Edward Island

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