Double circulant self-dual and LCD codes over Galois rings

Minjia Shi; Daitao Huang; Lin Sok; Patrick Solé

doi:10.3934/amc.2019011

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2019, Volume 13, Issue 1: 171-183. Doi: 10.3934/amc.2019011

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Double circulant self-dual and LCD codes over Galois rings

1.
School of Mathematical Sciences, Anhui University, Hefei 230601, China
2.
CNRS/LAGA, University of Paris, 2 rue de la Liberté, 93 526 Saint-Denis, France

* Corresponding author: Minjia Shi
* Corresponding author: Minjia Shi

Received: May 31, 2018

Published: December 2018

This paper is supported by National Natural Science Foundation of China (61672036), Excellent Youth Foundation of Natural Science Foundation of Anhui Province (1808085J20).

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This paper investigates the existence, enumeration, and asymptotic performance of self-dual and LCD double circulant codes over Galois rings of characteristic $p^2$ and order $p^4$ with $p$ an odd prime. When $p \equiv 3 \ ({\rm mod} \ 4),$ we give a method to construct a duality preserving bijective Gray map from such a Galois ring to $\mathbb{Z}_{p^2}^2.$ Closed formed enumeration formulas for double circulant codes that are self-dual (resp. LCD) are derived as a function of the length of these codes. Using random coding, we obtain families of asymptotically good self-dual and LCD codes over $\mathbb{Z}_{p^2}$ with respect to the metric induced by the standard ${\mathbb{F}}_p$ -valued Gray maps.

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Mathematics Subject Classification: Primary: 94B65; Secondary: 13K05, 13.95.

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References

[1]	A. Alahmadi, F. Özdemir and P. Solé, On self-dual double circulant codes, Des. Codes Cryptogr., 86 (2018), 1257-1265. doi: 10.1007/s10623-017-0393-x.
[2]	C. Carlet and S. Guilley, Complementary dual codes for counter-measures to side-channel attacks, Adv. in Math. of Comm., 10 (2016), 131-150. doi: 10.3934/amc.2016.10.131.
[3]	S. T. Dougherty, T. A. Gulliver and J. Wong, Self-Dual Codes over $ \mathbb{Z}_8$ and $ \mathbb{Z}_9$, Des. Codes Cryptogr., 41 (2006), 235-249. doi: 10.1007/s10623-006-9000-2.
[4]	S. T. Dougherty, J. L. Kim, B. Özkaya, L. Sok and P. Solé, The combinatorics of LCD codes: linear programming bound and orthogonal matrices, Int. J. of Information and Coding Theory, 4 (2017), 116-128. doi: 10.1504/IJICOT.2017.083827.
[5]	M. Harada and A. Munemasa, On the classification of self-dual $ \mathbb{Z}_k$-codes, Lecture Notes in Comput. Sci., Springer, 5921 (2009), 78-90. doi: 10.1007/978-3-642-10868-6_6.
[6]	W. C. Huffman and V. Pless, Fundamentals of Error Correcting Codes, Cambridge University Press, 2003. doi: 10.1017/CBO9780511807077.
[7]	K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, (2nd ed.), Springer, 1990. doi: 10.1007/978-1-4757-2103-4.
[8]	J.-L. Kim and Y. Lee, Construction of MDS self-dual codes over Galois rings, Des. Codes Cryptogr., 45 (2007), 247-258. doi: 10.1007/s10623-007-9117-y.
[9]	S. Ling and T. Blackford, $ \mathbb{Z}_{p^{k+1}}$-linear codes, IEEE Trans. Inf. Theory, 48 (2002), 2592-2605. doi: 10.1109/TIT.2002.801473.
[10]	S. Ling and P. Solé, On the algebraic structure of quasi-cyclic codes Ⅱ: Chain rings, Des. Codes Cryptogr., 30 (2003), 113-130. doi: 10.1023/A:1024715527805.
[11]	Y. Liu, M. Shi, Z. Sepasdar and P. Solé, Construction of Hermitian Self-dual constacyclic codes over $ {\mathbb{F}}_{q^2}+u{\mathbb{F}}_{q^2}$, Appl. and Comput. Math., 15 (2016), 359-369.
[12]	J. Massey, Linear codes with complementary duals, Discrete Math., 106/107 (1992), 337-342. doi: 10.1016/0012-365X(92)90563-U.
[13]	P. Moree, On primes in arithmetic progression having a prescribed primitive root, Journal of Number Theory, 78 (1999), 85-98. doi: 10.1006/jnth.1999.2409.
[14]	M. Shi, A. Alahmadi and P. Solé, Codes and Rings: Theory and Practice, Academic Press, 2017.
[15]	Z. X. Wan, Finite Fields and Galois Rings, World Scientific, 2003.