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2018, Volume 12, Issue 4: 723-739. Doi: 10.3934/amc.2018043

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A class of skew-cyclic codes over $\mathbb{Z}_4+u\mathbb{Z}_4$ with derivation

Amit Sharma^, and
Maheshanand Bhaintwal

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, 247667, India

* Corresponding author: Amit Sharma
* Corresponding author: Amit Sharma

Received: October 2017

Revised: March 2018

Published: November 2018

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Abstract

In this paper, we study a class of skew-cyclic codes using a skew polynomial ring over $R = \mathbb{Z}_4+u\mathbb{Z}_4;u^2 = 1$, with an automorphism $θ$ and a derivation $δ_θ$. We generalize the notion of cyclic codes to skew-cyclic codes with derivation, and call such codes as $δ_θ$-cyclic codes. Some properties of skew polynomial ring $R[x, θ, {δ_θ}]$ are presented. A $δ_θ$-cyclic code is proved to be a left $R[x, θ, {δ_θ}]$-submodule of $\frac{R[x, θ, {δ_θ}]}{\langle x^n-1 \rangle}$. The form of a parity-check matrix of a free $δ_θ$-cyclic codes of even length $n$ is presented. These codes are further generalized to double $δ_θ$-cyclic codes over $R$. We have obtained some new good codes over $\mathbb{Z}_4$ via Gray images and residue codes of these codes. The new codes obtained have been reported and added to the database of $\mathbb{Z}_4$-codes [2].

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Mathematics Subject Classification: Primary: 94B05, 94B15, 11T71; Secondary: 12D05.

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Table 1. *: = Existing good code [2,1], **: = New good code

	$C$	$\Phi(C)$	$Res(C)$	$C^*$
Set of generators	Code	$(n, 4^{k_1}2^{k_2}, d_L)$	$(n, 4^{k_1}2^{k_2}, d_L)$	$(n, 4^{k_1}2^{k_2}, d_L)$
$\{g_1(x), xg_1(x), x^2g_1(x)\}$	$C_1$	${(10, 4^6, 2)^{}}$	${(5, 4^42^1, 2)^{*}}$	$\mathbf{(10, 4^82^2, 2)}^{**}$
$\{g_2(x), xg_2(x), x^2g_2(x)\}$	$C_2$	${(20, 4^6, 8)}$	$(10, 4^6, 4)^*$	$(20, 4^{12}, 4)^*$
$\{g_3(x), xg_3(x), x^2g_3(x)\}$	$C_3$	${(20, 4^6, 6)}$	$(10, 4^5, 6)^*$	$(20, 4^{10}, 6)$
$\left\{ {{g_4}\left( x \right),x{g_4}\left( x \right),{x^2}{g_4}\left( x \right),{x^3}{g_4}\left( x \right)} \right\}$	$C_4$	${(24, 4^8, 6)}$	$(12, 4^8, 4)^*$	$(24, 4^{16}, 4)^*$
$\left\{ {{g_5}\left( x \right),x{g_5}\left( x \right),{x^2}{g_5}\left( x \right),{x^3}{g_5}\left( x \right)} \right\}$	$C_5$	${(28, 4^8, 6)}$	$(14, 4^8, 5)^*$	$(28, 4^{16}, 5)^*$
$\left\{ {{g_6}\left( x \right),x{g_6}\left( x \right),{x^2}{g_6}\left( x \right),{x^3}{g_6}\left( x \right)} \right\}$	$C_6$	${(30, 4^8, 6)}$	$(15, 4^8, 6)^*$	$(30, 4^{16}, 6)$
$\left\{ {{g_7}\left( x \right),x{g_7}\left( x \right),{x^2}{g_7}\left( x \right),{x^3}{g_7}\left( x \right)} \right\}$	$C_7$	${(36, 4^8, 8)}$	$(18, 4^8, 8)^*$	$(36, 4^{16}, 8)^*$

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Table 2. $^*$: = Existing good code [2,1], $^{**}$: = New good code;

	$C$	$\Phi(C)$	$Res(C)$	$C^*$
Set of generators	Name	$(n, M, d_L)$	$(n, 4^{k_1}2^{k_2}, d_L)$	$(n, 4^{k_1}2^{k_2}, d_L)$
$\{h_0(x), xh_1(x)\}$	$A_1$	${(10,128, 2)^{}}$	${(5, 4^32^1, 2)^{*}}$	$(10, 4^62^2, 2)$
$\{h_1(x), xh_1(x), x^2h_1(x)\}$	$A_2$	${(12, 4096, 2)^{}}$	${(6, 4^52^1, 2)^{*}}$	$\mathbf{(12, 4^{10}2^2, 2)}^{**}$
$\{h_2(x), xh_2(x), x^2h_2(x), x^3h_2(x)\}$	$A_3$	${(14, 65536, 2)^{}}$	${(7, 4^62^1, 2)^{*}}$	${(14, 4^{12}2^2, 2)}$
$\{h_3(x), xh_3(x), x^3h_2(x), x^3h_3(x)\}$	$A_3$	${(16, 65536, 4)^{}}$	$\mathbf{(8, 4^7, 2)^{**}}$	$\mathbf{(16, 4^{14}, 2)}^{**}$

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