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: June 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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