# American Institute of Mathematical Sciences

February  2019, 13(1): 171-183. doi: 10.3934/amc.2019011

## 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

Received  June 2018 Published  December 2018

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

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.

Citation: Minjia Shi, Daitao Huang, Lin Sok, Patrick Solé. Double circulant self-dual and LCD codes over Galois rings. Advances in Mathematics of Communications, 2019, 13 (1) : 171-183. doi: 10.3934/amc.2019011
##### 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [6] W. C. Huffman and V. Pless, Fundamentals of Error Correcting Codes, Cambridge University Press, 2003. doi: 10.1017/CBO9780511807077. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [12] J. Massey, Linear codes with complementary duals, Discrete Math., 106/107 (1992), 337-342. doi: 10.1016/0012-365X(92)90563-U. Google Scholar [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. Google Scholar [14] M. Shi, A. Alahmadi and P. Solé, Codes and Rings: Theory and Practice, Academic Press, 2017. Google Scholar [15] Z. X. Wan, Finite Fields and Galois Rings, World Scientific, 2003. Google Scholar

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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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [6] W. C. Huffman and V. Pless, Fundamentals of Error Correcting Codes, Cambridge University Press, 2003. doi: 10.1017/CBO9780511807077. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [12] J. Massey, Linear codes with complementary duals, Discrete Math., 106/107 (1992), 337-342. doi: 10.1016/0012-365X(92)90563-U. Google Scholar [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. Google Scholar [14] M. Shi, A. Alahmadi and P. Solé, Codes and Rings: Theory and Practice, Academic Press, 2017. Google Scholar [15] Z. X. Wan, Finite Fields and Galois Rings, World Scientific, 2003. Google Scholar
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