Skew constacyclic codes over the local Frobenius non-chain rings of order 16

Nuh Aydin; Yasemin Cengellenmis; Abdullah Dertli; Steven T. Dougherty; Esengül Saltürk

doi:10.3934/amc.2020005

Article Contents

2020, Volume 14, Issue 1: 53-67. Doi: 10.3934/amc.2020005

This issue Previous Article Some generalizations of good integers and their applications in the study of self-dual negacyclic codes Next Article A complete classification of partial MDS (maximally recoverable) codes with one global parity

Skew constacyclic codes over the local Frobenius non-chain rings of order 16

1.
Department of Mathematics, Kenyon College, Gambier, OH 43022, USA
2.
Department of Mathematics, Trakya University, 22030 Edirne, Turkey
3.
Department of Mathematics, Ondokuz Mayis University, 55139 Samsun, Turkey
4.
Department of Mathematics, University of Scranton, Scranton, PA. 18518, USA
5.
The Scientific and Technological, Research Council Of Turkey, 41401 Kocaeli, Turkey

^* Corresponding author: Steven T. Dougherty
^* Corresponding author: Steven T. Dougherty

Esengül Saltürk would like to thank TUBITAK (The Scientific and Technological Research Council of Turkey) for their support while writing this paper

Received: June 2018

Revised: December 2018

Published: August 2019

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Abstract

We introduce skew constacyclic codes over the local Frobenius non-chain rings of order 16 by defining non-trivial automorphisms on these rings. We study the Gray images of these codes, obtaining a number of binary and quaternary codes with good parameters as images of skew cyclic codes over some of these rings.

Keywords:

Mathematics Subject Classification: Primary: 11T71; Secondary: 94B15, 13H99.

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Table 1. Optimal binary linear codes

n	g(x) or h(x)	Binary Parameters
$ 8 $	$ g=x^2 + ux + 1 $	$ [32,24,4] $
$ 8 $	$ g=x^4+(uv+u)x^3+(v+u)x^2+ux+1 $	$ [32,16,8] $
$ 6 $	$ h=x^2+(u+v+1)x + 1 $	$ [24,8,8] $
$ 6 $	$ h=x+uv+1 $	$ [24,4,12] $

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Table 2. Best-known binary linear codes

n	g(x) or h(x)	Binary Parameters
$ 12 $	$ h=x^5+(u+v+1)x^4+x^3+x^2+(uv+v+1)x+v+1 $	$ [48,20,12] $
$ 12 $	$ g=x^4+(uv+u+1)x^3+x+1 $	$ [48,32,6] $
$ 12 $	$ g=x^5+ux^4+x^3+ (u+v+ 1)x^2+v+1 $	$ [48,28,8] $
$ 16 $	$ h=x^4+vx^3+ux^2+1 $	$ [64,24,16] $

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Table 3. New binary QC codes

n	g(x) or h(x)	Binary Parameters
$ 8 $	$ g=x^3+(uv+1)x^2+x+1 $	$ [32,20,4] $
$ 10 $	$ g=x^4+(u+v+ 1)x^3+x^2+x+1 $	$ [40,24,4] $
$ 12 $	$ g=x^3+ux^2+1 $	$ [48,36,4] $
$ 14 $	$ h=x^5+(uv+1)x^4+x^3+uvx^2+1 $	$ [56,20,7] $
$ 14 $	$ h=x^4+(u+v+1)x^3+x^2+1 $	$ [56,16,12] $
$ 14 $	$ g=x^3+ (u+v+1)x^2+1 $	$ [56,44,3] $
$ 14 $	$ h=x^3+ (u+v+1)x^2+1 $	$ [56,12,16] $
$ 16 $	$ h=x^6+(uv+v+1)x^4+ux^3+x^2+vx+uv+1 $	$ [64,24,16] $
$ 20 $	$ h=x^4+x^3+(u+1)x^2+x+u+v+1 $	$ [80,16,28] $
$ 18 $	$ h=x^4+(u+v+1)x^3+(u+v+1)x+1 $	$ [72,16,9] $
$ 28 $	$ h=x+u+v+1 $	$ [112,4,56] $
$ 28 $	$ h=x^4+x^2+(u+1)x+(u+1)v+1 $	$ [112,16,40] $
$ 30 $	$ h= x+uv+1 $	$ [120,4,60] $
$ 32 $	$ h=x+u+v+1 $	$ [128,4,64] $
$ 32 $	$ h=x^3+(u+1)x^2+x+1 $	$ [128,12,32] $

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Table 4. New quaternary codes

n	g(x) or h(x)	$ {\mathbb{Z}}_4 $ Parameters
$ 8 $	$ g=x^3+(u+1)x^2+3x+u+1 $	$ [24,10,9] $
$ 8 $	$ g=x^2+(3u+2)x+3u+3 $	$ [24,12,7] $
$ 8 $	$ g=x+u+1 $	$ [16,14,2] $
$ 12 $	$ g=x^2+(u+3)x+1 $	$ [36,20,8] $
$ 12 $	$ h=x^4+(3u+1)x^3+(u+2)x^2+(3u+3)x+3u +3 $	$ [24,8,12] $
$ 12 $	$ h=x^4+(3u+3)x^3+(u+2)x^2+(u+3)x+u+1 $	$ [36,8,20] $
$ 12 $	$ h=x^5+(2u+1)x^4+3x^3+(2u+1)x^2+3x+1 $	$ [24,10,8] $
$ 12 $	$ h=x^5+2ux^4+(3u+3)x^3+(2u+3)x^2+(3u+2)x+3 $	$ [36,10,17] $
$ 12 $	$ g=x^5+3ux^4+x^3+(3u+1)x^2+1 $	$ [36,14,13] $
$ 14 $	$ h=x^4+(3u+3)x^3+(u+3)x^2+ux+u+1 $	$ [28,8,18] $
$ 14 $	$ h=x^4+(3u+3)x^3+(u+3)x^2+ux+u+1 $	$ [42,8,26] $
$ 14 $	$ h=x^3+(u+3)x^2+(3u+2)x+u+1 $	$ [28,6,18] $
$ 14 $	$ h=x^3+(u+3)x^2+(3u+2)x+u+1 $	$ [42,6,29] $
$ 14 $	$ g=x^4+(u+3)x^3 + x^2 + (3u + 2)x + 2u+1 $	$ [28,20,6] $
$ 14 $	$ g=x^4+(u+3)x^3 + x^2 + (3u + 2)x + 2u+1 $	$ [42,20,13] $
$ 16 $	$ g=x^4 + (u + 2)x^3 + ux^2 + 2x + 1 $	$ [48,24,13] $
$ 16 $	$ g=x^3+(u+3)x^2+(3u+1)x+3u+3 $	$ [32,26,4] $
$ 16 $	$ g=x^3+(u+3)x^2+(3u+1)x+3u+3 $	$ [48,26,11] $
$ 16 $	$ g=x^2+(u+2)x+1 $	$ [32,28,2] $
$ 16 $	$ g=x^2+(u+2)x+1 $	$ [48,28,9] $
$ 18 $	$ g=x^4 + (2u + 3)x^3 + ux^2 + (2u + 1)x + 3u + 3 $	$ [54,28,13] $
$ 18 $	$ g=x^3 + ux^2 + 2x + 1 $	$ [54,30,10] $
$ 18 $	$ h=x^3+(3u+2)x^2+3ux +2u + 3 $	$ [54,6,34] $
$ 20 $	$ g=x^4 + 3x^3 + (3u + 1)x^2 + (3u + 1)x + 1 $	$ [60,32,13] $
$ 20 $	$ g=x^4 + (u + 3)x^3 + x^2 + x + 1 $	$ [40,32,4] $
$ 20 $	$ g=x^3 + (3u + 3)x^2 + (2u + 1)x + u + 3 $	$ [60,34,12] $
$ 24 $	$ g=x^3 + 3ux^2 + (u + 2)x + 1 $	$ [48,42,4] $
$ 24 $	$ g=x^3 + 3ux^2 + (u + 2)x + 1 $	$ [72,42,10] $
$ 30 $	$ g=x^4 + (u + 1)x^3 + 1 $	$ [60,52,3] $
$ 30 $	$ g=x^4 + (u + 1)x^3 + 1 $	$ [90,52,8] $
$ 32 $	$ g=x + u + 1 $	$ [64,62,2] $
$ 32 $	$ g=x + u + 1 $	$ [96,62,5] $
$ 32 $	$ h=x^4 + ux^2 + (u + 2)x + 3u + 1 $	$ [96,8,60] $

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