Quaternary group ring codes: Ranks, kernels and self-dual codes

Steven T. Dougherty; Cristina Fernández-Córdoba; Roger Ten-Valls; Bahattin Yildiz

doi:10.3934/amc.2020023

Article Contents

2020, Volume 14, Issue 2: 319-332. Doi: 10.3934/amc.2020023

This issue Previous Article Algebraic dependence in generating functions and expansion complexity Next Article Linear programming bounds for distributed storage codes

Quaternary group ring codes: Ranks, kernels and self-dual codes

1.
Department of Mathematics, University of Scranton, Scranton, PA 18510, USA
2.
Department of Information and Communications Engineering, Universitat Autònoma de Barcelona, 08193-Bellaterra, Spain
3.
Department of Mathematics and Statistics, Northern Arizona University, Flagstaff, AZ 86001, USA

Received: June 30, 2018

Early access: September 2019

Published: May 2020

This work has been partially supported by the Spanish MINECO under Grant TIN2016-77918-P (AEI/FEDER, UE)

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Abstract

We study $ G $-codes over the ring $ {\mathbb{Z}}_4 $, which are codes that are held invariant by the action of an arbitrary group $ G $. We view these codes as ideals in a group ring and we study the rank and kernel of these codes. We use the rank and kernel to study the image of these codes under the Gray map. We study the specific case when the group is the dihedral group and the dicyclic group. Finally, we study quaternary self-dual dihedral and dicyclic codes, tabulating the many good self-dual quaternary codes obtained via these constructions, including the octacode.

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Mathematics Subject Classification: Primary: 16S34; Secondary: 94B15.

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Table 1. (Extremal) Dihedral Self-dual Codes of length 4

First Row of A	First row of B	Min Lee Weight	Lee Weight Distribution
(1, 1)	(1, 3)	4	$ 1+14z^4+z^{8} $
(1, 1)	(3, 1)	4	$ 1+14z^4+z^{8} $
(1, 3)	(1, 1)	4	$ 1+14z^4+z^{8} $
(1, 3)	(3, 3)	4	$ 1+14z^4+z^{8} $
(3, 1)	(1, 1)	4	$ 1+14z^4+z^{8} $
(3, 1)	(3, 3)	4	$ 1+14z^4+z^{8} $
(1, 3)	(1, 3)	4	$ 1+14z^4+z^{8} $
(3, 3)	(3, 1)	4	$ 1+14z^4+z^{8} $

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Table 2. (Extremal) Dihedral Self-dual Codes of length 8

First Row of $ A $	First row of $ B $	Min Lee Weight	Lee Weight Distribution
(0, 0, 2, 2)	(1, 1, 3, 1)	4	$ 1+28z^4+198z^{8}+\dots $
(0, 0, 0, 0)	(1, 3, 1, 1)	4	$ 1+28z^4+198z^{8}+\dots $
(0, 0, 0, 2)	(3, 1, 3, 1)	4	$ 1+12z^4+64z^{6}+102z^8+\dots $
(0, 0, 2, 0)	(1, 1, 3, 3)	4	$ 1+12z^4+64z^{6}+102z^8+\dots $

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Table 3. Best Dicyclic Self-dual Codes of lengths 4, 8 and 12

$ n $	1st row of $ A $	1st row of $ B $	1st row of $ C $	Min Lee Weight	Gray Image Linear
$ 4 $	(1, 3)	(3, 3)	(3, 3)	4	Yes
$ 8 $	(0, 0, 0, 2)	(3, 3, 3, 3)	(3, 3, 3, 3)	4	Yes
$ 8 $	(0, 0, 1, 1)	(0, 0, 1, 3)	(1, 3, 0, 0)	4	No
$ 8 $	(0, 0, 1, 1)	(0, 1, 1, 2)	(1, 2, 0, 1)	$ 6^* $	No
$ 12 $	(0, 0, 0, 0, 0, 0)	(0, 1, 3, 0, 1, 1)	(0, 1, 1, 0, 1, 3)	4	Yes

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