Constacyclic and quasi-twisted Hermitian self-dual codes over finite fields

Ekkasit Sangwisut; Somphong Jitman; Patanee Udomkavanich

doi:10.3934/amc.2017045

Article Contents

2017, Volume 11, Issue 3: 595-613. Doi: 10.3934/amc.2017045

This issue Previous Article Network encoding complexity: Exact values, bounds, and inequalities Next Article Finite nonassociative algebras obtained from skew polynomials and possible applications to (f, σ, δ)-codes

Constacyclic and quasi-twisted Hermitian self-dual codes over finite fields

1.
Department of Mathematics and Statistics, Faculty of Science, Thaksin University, Phatthalung Campus, Phatthalung 93110, Thailand
2.
Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom 73000, Thailand
3.
Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand

* Corresponding author
* Corresponding author

Received: November 2015

Published: September 2017

S. Jitman was supported by the Thailand Research Fund under Research Grant TRG5780065.

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Abstract

Constacyclic and quasi-twisted Hermitian self-dual codes over finite fields are studied. An alternative algorithm for factorizing $x^n-\lambda $ over ${\mathbb{F}_{{q^2}}}$ is given, where $λ$ is a unit in ${\mathbb{F}_{{q^2}}}$. Based on this factorization, the dimensions of the Hermitian hulls of $\lambda $-constacyclic codes of length $n$ over ${\mathbb{F}_{{q^2}}}$ are determined. The characterization and enumeration of constacyclic Hermitian self-dual (resp., complementary dual) codes of length $n$ over ${\mathbb{F}_{{q^2}}}$ are given through their Hermitian hulls. Subsequently, a new family of MDS constacyclic Hermitian self-dual codes over ${\mathbb{F}_{{q^2}}}$ is introduced.

As a generalization of constacyclic codes, quasi-twisted Hermitian self-dual codes are studied. Using the factorization of $x^n-\lambda $ and the Chinese Remainder Theorem, quasi-twisted codes can be viewed as a product of linear codes of shorter length over some extension fields of ${\mathbb{F}_{{q^2}}}$. Necessary and sufficient conditions for quasi-twisted codes to be Hermitian self-dual are given. The enumeration of such self-dual codes is determined as well.

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Mathematics Subject Classification: Primary: 94B15, 11T06; Secondary: 94B05.

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Table 1. MDS constacyclic Hermitian self-dual codes over $\mathbb{F}_{q^2}$

$q$	$m$	$i$	Parameters	$T$
$3$	$2$	$1$	$[4,2,3]$	$\{1, 3\}$
$5$	$1$	$1$	$[6,3,4]$	$\{1, 3, 5\}$
$7$	$3$	$1$	$[8,4,5]$	$\{1, 3, 5, 7\}$
	$3$	$2$	$[4,2,3]$	$\{1, 5\}$
$9$	$1$	$1$	$[10,5,6]$	$\{1, 3, 5, 7, 9\}$
$11$	$2$	$1$	$[12,6,7]$	$\{1, 3, 5, 7, 9, 11\}$
	$2$	$2$	$[6,3,4]$	$\{1, 5, 9\}$
$13$	$1$	$1$	$[14,7,8]$	$\{1, 3, 5, 7, 9, 11, 13\}$

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