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May 2018, 12(2): 363-385. doi: 10.3934/amc.2018023
The weight distribution of quasi-quadratic residue codes
Nigel Boston 1, and Jing Hao 2,
Department of Mathematics, Department of Electric and Computer Engineering, University of Wisconsin-Madison, WI 53706, United States
Department of Mathematics, University of Wisconsin-Madison, WI 53706, United States
Received May 2017 Published March 2018
Fund Project: The first author is supported by Simons Foundation grant MSN179747
Figure(1) / Table(4)
We investigate a family of codes called quasi-quadratic residue (QQR) codes. We are interested in these codes mainly for two reasons: Firstly, they have close relations with hyperelliptic curves and Goppa's Conjecture, and serve as a strong tool in studying those objects. Secondly, they are very good codes. Computational results show they have large minimum distances when $p\equiv 3 \pmod 8$.
Our studies focus on the weight distributions of these codes. We will prove a new discovery about their weight polynomials, i.e. they are divisible by $(x^2 + y^2)^{d-1}$, where $d$ is the corresponding minimum distance. We also show that the weight distributions of these codes are asymptotically normal. Based on the relation between QQR codes and hyperelliptic curves, we will also prove a result on the point distribution on hyperelliptic curves.
Keywords: Algebraic coding theory, weight enumerator, automorphism group, shadow, moment, quadratic residue code, hyperelliptic curve.
Mathematics Subject Classification: Primary: 94B15, 94B60; Secondary: 11G20.
Citation: Nigel Boston, Jing Hao. The weight distribution of quasi-quadratic residue codes. Advances in Mathematics of Communications, 2018, 12 (2) : 363-385. doi: 10.3934/amc.2018023
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Figure 1. distribution comparison
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Table 1. Computational Results
$p$ $d$ $\delta$ Divisible by
3 2 0.33 $(x^2 + y^2)^3$
11 6 0.27 $(x^2 + y^2)^7$
43 14 0.16 $(x^2 + y^2)^{15}$
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Table 2. Weight polynomials posted on [18]
$p$ $k$ $d$ Divisible by
89 45 17 $(x+y)^{17}$
103 52 19 $(x+y)^{19}$
113 57 15 $(x+y)$
151 76 19 $(x+y) $
Table 3. Weight polynomials in references
Table 4. Correction for $p = 127$
$i$ $A_i$ in table $A_i$ corrected
51 223367511592873280 223367511592873284
56 1080334587116677120 1080334587116677140
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Nigel Boston Jing Hao