$ \mathbb{Z}_{4}\mathbb{Z}_{4}[u] $-additive cyclic and constacyclic codes

Habibul Islam; Om Prakash; Patrick Solé

doi:10.3934/amc.2020094

Article Contents

2021, Volume 15, Issue 4: 737-755. Doi: 10.3934/amc.2020094

This issue Previous Article A generic construction of rotation symmetric bent functions Next Article Several new classes of (balanced) Boolean functions with few Walsh transform values

$ \mathbb{Z}_{4}\mathbb{Z}_{4}[u] $-additive cyclic and constacyclic codes

1.
Department of Mathematics, Indian Institute of Technology Patna, Patna- 801 106, India
2.
I2M, (CNRS, Aix-Marseille University, Centrale Marseille), Marseilles, France

^* Corresponding author: Om Prakash
^* Corresponding author: Om Prakash

Received: December 31, 2019

Revised: April 30, 2020

Early access: July 2020

Published: November 2021

The research is supported by the University Grants Commission (UGC), Govt. of India

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Abstract

We study mixed alphabet cyclic and constacyclic codes over the two alphabets $ \mathbb{Z}_{4}, $ the ring of integers modulo $ 4 $, and its quadratic extension $ \mathbb{Z}_{4}[u] = \mathbb{Z}_{4}+u\mathbb{Z}_{4}, u^{2} = 0. $ Their generator polynomials and minimal spanning sets are obtained. Further, under new Gray maps, we find cyclic, quasi-cyclic codes over $ \mathbb{Z}_{4} $ as the Gray images of both $ \lambda $-constacyclic and skew $ \lambda $-constacyclic codes over $ \mathbb{Z}_{4}[u] $. Moreover, it is proved that the Gray images of $ \mathbb{Z}_{4}\mathbb{Z}_{4}[u] $-additive constacyclic and skew $ \mathbb{Z}_{4}\mathbb{Z}_{4}[u] $-additive constacyclic codes are generalized quasi-cyclic codes over $ \mathbb{Z}_{4} $. Finally, several new quaternary linear codes are obtained from these cyclic and constacyclic codes.

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Mathematics Subject Classification: 94B15, 94B05, 94B60.

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Table 1. Gray images of $(3+2u)$-constacyclic codes over $\mathbb Z_4[u]$

$\beta$	$h(x)$	$k(x)$	$\phi_{1}(\mathcal{S})$	$\phi_{2}(\mathcal{S})$
$3$	$[1, 3+2u, 3+u]$	$[3u, u]$	$[6, 4^0 2^4, 2]^*$	$[6, 4^4 2^0, 2]^{\#}$
$7$	$[1, 3+2u, 3, 0, 1+u]$	$[3u, 3u, 2u, u]$	$[14, 4^32^7, 2]$	$[14, 4^72^3, 4]^{\#}$
$7$	$[3+2u, 2, 3+2u, 1+u]$	$[3u, 3u, 2u, u]$	$[14, 4^0 2^{11}, 2]$	$[14, 4^7 2^4, 2]$
$9$	$[3+2u, 0, 2, 1+u]$	$[u, 0, 0, 3u, 0, 0, 3u]$	$[18, 4^02^{15}, 2]$	$[18, 4^9 2^6, 2]$
$9$	$[3+2u, 3, 0, 1, 1+2u, 0, 1+2u, 1+u]$	$[u, 3u, 3u]$	$[18, 4^0 2^{11}, 2]^*$	$[18, 4^9 2^2, 2]$
$15$	$[3+2u, 3, 2, 1, 2+u]$	$[u, 2u, 2u, 3u, u, u, 3u]$	$[30, 4^0 2^{26}, 2]$	$[30, 4^{15} 2^{11}, 2]^*$
$15$	$[1, 0, 2, 2, 1, 3+2u, 1, 3+2u, 3+u]$	$[u, 3u, 3u]$	$[30, 4^0 2^{20}, 4]$	$[30, 4^{15} 2^5, 4]^*$
$17$	$[3+2u, 3, 2, 1, 2, 2, 1+2u, 2, 1+2u, 2, 1+2u, 1+u]$	$[u, 3u, 3u, 0, 3u, 0, 3u, 3u, 3u]$	$[34, 4^0 2^{25}, 2]^*$	$[34, 4^{17} 2^8, 2]^*$
$17$	$[1, 0, 2, 1+2u, 1, 1+2u, 2, 0, 3+u]$	$[u, 3u, 3u, 0, 3u, 0, 3u, 3u, 3u]$	$[34, 4^0 2^{26}, 2]^*$	$[34, 4^{17} 2^9, 2]^*$
$21$	$[3+2u, 0, 0, 3, 0, 0, 0, 0, 2, 1+u]$	$[u, 3u, 3u, 0, 3u, 2u, 3u]$	$[42, 4^0 2^{33}, 2]^*$	$[42, 4^{21} 2^{18}, 2]^*$
$21$	$[3+2u, 3, 2, 2, 3+2u, 1, 2, 1, 3+2u, 3+u]$	$[3u, 0, 2u, u]$	$[42, 4^0 2^{33}, 2]^*$	$[42, 4^{21} 2^{12}, 2]^*$

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Table 2. $ \mathbb{Z}_4 $-Gray images of $ \mathbb{Z}_4\mathbb{Z}_4[u] $-additive cyclic codes of length $ (\alpha, \beta) $

$ (\alpha, \beta) $	Generators	$ \Phi_{1}(\mathcal{C}) $	$ \Phi_{2}(\mathcal{C}) $
$ (3, 3) $	$ g_1=g_2=x^2+x+1, g_3=x+1 $, $ a_1=a_2=a_3=1, p=1, f_1=f_2=x+1 $	$ [9, 4^3 2^3, 1]^* $	$ [9, 4^6 2^0, 1] $
$ (3, 3) $	$ g_1=g_2=g_3=x^3-1, a_1=x^2+x+1 $, $ a_2=a_3=x+3, p=1, f_1=f_2=x^2+2 $	$ [9, 4^3 2^2, 1]^* $	$ [9, 4^5 2^1, 1]^{*} $
$ (3, 7) $	$ g_1=x^3-1, g_2=x^4+x^3+3x^2+2x+1, g_3=x^7-1 $,
	$ a_1=x^2+x+1, a_2=x+3, a_3=x^3+3x^2+2x+3, p=1, f_1=f_2=x+1 $	$ [17, 4^2 2^8, 2]^* $	$ [17, 4^9 2^1, 2]^* $
$ (7, 3) $	$ g_1=x^7-1, g_2=x^3-1, g_3=x+3 $,
	$ a_1=x^4+2x^3+3x^2+x+1, a_2=x^2+x+1, a_3=1, p=1, f_1=f_2=x+3 $	$ [13, 4^3 2^5, 2]^* $	$ [13, 4^6 2^2, 2]^* $
$ (7, 7) $	$ g_1=x^4+x^3+3x^2+2x+1, g_2=x^4+2x^3+3x^2+x+1, g_3=x+3 $,
	$ a_1=x^3+2x^2+x+3, a_2=x^3+3x^2+2x+3, a_3=1, p=1, f_1=f_2=x+3 $	$ [21, 4^6 2^8, 2]^* $	$ [21, 4^{13} 2^2, 2]^* $
$ (9, 9) $	$ g_1=x^2+x+1, g_2=x^7+3x^6+x^4+3x^3+x+3, g_3=x^6+x^3+1 $,
	$ a_1=1, a_2=x^6+x^3+1, a_3=1, p=1, f_1=f_2=x+3 $	$ [27, 4^9 2^9, 2]^* $	$ [27, 4^{16} 2^4, 2]^* $
$ (3, 9) $	$ g_1=x^2+x+1, g_2=x^7+3x^6+x^4+3x^3+x+3, g_3=x^3+3 $,
	$ a_1=1, a_2=a_3=x+3, p=1, f_1=f_2=x+3 $	$ [21, 4^3 2^9, 1]^* $	$ [21, 4^{12} 2^0, 1] $
$ (9, 3) $	$ g_1=x^3+3, g_2=x^3-1, g_3=x+3 $,
	$ a_1=x^2+x+1, a_2=x+3, a_3=1, p=1, f_1=f_2=x+3 $	$ [15, 4^8 2^4, 2]^* $	$ [15, 4^{11} 2^1, 2]^* $
$ (7, 9) $	$ g_1=x^4+x^3+3x^2+2x+1, g_2=x^7+3x^6+x^4+3x^3+x+3, g_3=x^2+x+1 $,
	$ a_1=x^3+2x^2+x+3, a_2=x^6+x^3+1, a_3=1, p=1, f_1=f_2=x+1 $	$ [25, 4^6 2^{10}, 2]^* $	$ [25, 4^{15} 2^1, 2]^* $

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