Constacyclic and quasi-twisted codes over $ \mathbb{Z}_{q}[u]/\langle u^{2}-1\rangle $ and new $ \mathbb{Z}_4 $-linear codes

Amina Bellil; Kenza Guenda; Nuh Aydin; Peihan Liu; T. Aaron Gulliver

doi:10.3934/amc.2023026

Article Contents

2024, Volume 18, Issue 6: 1877-1892. Doi: 10.3934/amc.2023026

This issue Previous Article Tight security analysis of the public Permutation-based $ {{\textsf{PMAC_Plus}}} $ Next Article The $ b $-weight distribution for MDS codes

Constacyclic and quasi-twisted codes over $ \mathbb{Z}_{q}[u]/\langle u^{2}-1\rangle $ and new $ \mathbb{Z}_4 $-linear codes

1.
Faculty of Mathematics, USTHB, Algiers, Algeria
2.
Department of Mathematics and Statistics, Kenyon College, Gambier, Ohio, 43022 USA
3.
Paulson School of Engineering and Applied Sciences, Harvard University, Boston, MA, 02134 USA
4.
Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC, Canada

^*Corresponding author: Nuh Aydin
^*Corresponding author: Nuh Aydin

Received: January 2023

Revised: May 2023

Early access: July 31, 2023

Published online: July 31, 2023

The third author is supported by a Fulbright scholarship.

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Abstract

In this paper, we investigate the algebraic structures and properties of constacyclic and quasi-twisted (QT) codes over the ring $ R = \mathbb{Z}_{q}+u\mathbb{Z}_{q} $ with $ u^{2} = 1 $. We show that the image of a constacyclic code over $ R $ under a natural Gray map is a QT code of index $ 2 $ over $ \mathbb{Z}_q $. Given the decomposition of a QT code, we find the decomposition of its dual code. We present 116 new linear codes over $ \mathbb{Z}_{4} $ from the Gray images of QT codes over this ring with $ q = 4 $. Finally, a characterization of linear complementary pair (LCP) constacyclic codes over $ R $ is provided.

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Mathematics Subject Classification: Primary: 94B60; Secondary: 94B65.

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Table 1. Examples of new $ \mathbb{Z}_4 $ codes $ [n,k_1,k_2,d] $ obtained by Gray maps from 2-QT codes of the form $ \langle gf_1,gf_2 \rangle $ over $ \mathbb{Z}_4+u\mathbb{Z}_4 $

$ [n,k,d]_2 $	$ x^m-\lambda $	$ g $	$ f_1 $	$ f_2 $
$ [16,2,1,8] $	$ x^4 + 3 $	$ x^2 + 2u + 1 $	$ x + 2u + 1 $	$ x + 2u + 1 $
$ [ 24, 2, 2, 12 ] $	$ x^6 + 1 $	$ x^4 + 3x^2 + 1 $	$ x + u + 2 $	$ x+u $
$ [ 24, 4, 6, 8 ] $	$ x^6 + 3 $	$ x + u + 2 $	$ x^4 + (u + 3)x^2 + (u + 2)x + u + 3 $	$ x^4 + 3ux^3 + (2u + 1)x + u + 2 $
$ [ 24, 8, 2, 8 ] $	$ x^6 + 2 $	$ x + u + 2 $	$ x^4 + ux^3 + (u + 2)x^2 + (3u + 3)x + 2u + 1 $	$ x^4 + (3u + 1)x^3 + (u + 2)x^2 + (u + 3)x + 3u + 3 $
$ [ 28, 6, 2, 12 ] $	$ x^7 + 3u $	$ x^3 + 2ux^2 + x + 3u $	$ x^3 + (2u + 2)x^2 + (u + 2)x + u + 3 $	$ x^3 + (3u + 1)x^2 + (2u + 2)x + 1 $
$ [ 32, 2, 6, 16 ] $	$ x^8 + 3 $	$ x^4 + 2ux^2 + (2u + 2)x + 2u + 3 $	$ x^3 + 3ux^2 + (2u + 3)x + u + 2 $	$ x^3 + ux^2 + 3x + 3u + 2 $
$ [ 36, 12, 4, 10 ] $	$ x^9 + u $	$ x+u $	$ x^7 + (3u + 2)x^6 + (u + 2)x^5 + x^4 + 2ux^3 + 2x^2 + (2u + 1)x + u + 1 $	$ x^7 + 2x^6 + (2u + 1)x^5 + (u + 3)x^4 + (u + 1)x^3 + (3u + 3)x^2 + 2ux + u + 2 $

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