Reversible $ G $-codes over the ring $ {\mathcal{F}}_{j,k} $ with applications to DNA codes

Yasemin Cengellenmis; Abdullah Dertli; Steven T. Dougherty; Adrian Korban; Serap Şahinkaya; Deniz Ustun

doi:10.3934/amc.2021056

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2021, Volume 0. Doi: 10.3934/amc.2021056

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Reversible $ G $-codes over the ring $ {\mathcal{F}}_{j,k} $ with applications to DNA codes

1.
Faculty of Science, Department of Mathematics, Trakya University, Edirne, Turkey
2.
Science and Art Faculty, Department of Mathematics, Ondokuz Mays University, Samsun, Turkey
3.
Department of Mathematics, University of Scranton, Scranton, USA
4.
Department of Mathematical and Physical Sciences, University of Chester, Chester, England
5.
Faculty of Engineering, Department of Natural and Mathematical Sciences, Tarsus University, Mersin, Turkey
6.
Faculty of Engineering, Department of Computer Engineering, Tarsus University, Mersin, Turkey

^* Corresponding author: Adrian Korban
^* Corresponding author: Adrian Korban

Received: February 28, 2021

Revised: August 31, 2021

Early access: December 2021

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Abstract

In this paper, we show that one can construct a $ G $-code from group rings that is reversible. Specifically, we show that given a group with a subgroup of order half the order of the ambient group with an element that is its own inverse outside the subgroup, we can give an ordering of the group elements for which $ G $-codes are reversible of index $ \alpha $. Additionally, we introduce a new family of rings, $ {\mathcal{F}}_{j,k} $, whose base is the finite field of order $ 4 $ and study reversible $ G $-codes over this family of rings. Moreover, we present some possible applications of reversible $ G $-codes over $ {\mathcal{F}}_{j,k} $ to reversible DNA codes. We construct many reversible $ G $-codes over $ {\mathbb{F}}_4 $ of which some are optimal. These codes can be used to obtain reversible DNA codes.

Keywords:

Mathematics Subject Classification: Primary: 94B05; Secondary: 16S34.

Citation:

FullText(HTML)

Table 1. Reversible Group Codes from $ {\mathcal{G}}_1 $ over $ {\mathbb{F}}_4 $

$ C_i $	$ [n,k,d] $
$ C_1 $	$ [24,11,6] $
$ C_2 $	$ [24,10,8] $
$ C_3 $	$ [26,18,4] $
$ C_4 $	$ [28,16,4] $
$ C_5 $	$ [28,17,4] $
$ C_6 $	$ [28,18,4] $
$ C_7 $	$ [28,19,4] $
$ C_8 $	$ [28,20,4] $
$ C_9 $	$ [30,18,6] $
$ C_{10} $	$ [30,19,6] $
$ C_{11} $	$ {\bf{[30,20,6]}} $
$ C_{12} $	$ [32,18,4] $
$ C_{13} $	$ [32,20,4] $
$ C_{14} $	$ [32,22,4] $

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Table 2. Reversible Group Codes from $ {\mathcal{G}}_2 $ over $ {\mathbb{F}}_4 $

$ C_i $	$ [n,k,d] $
$ C_{15} $	$ [40,24,5] $
$ C_{16} $	$ [40,28,4] $
$ C_{17} $	$ [40,28,5] $
$ C_{18} $	$ [40,28,6] $
$ C_{19} $	$ [40,29,4] $
$ C_{20} $	$ [40,30,5] $
$ C_{21} $	$ [44,34,2] $
$ C_{22} $	$ [44,32,4] $
$ C_{23} $	$ {\bf{[46,44,2]}} $
$ C_{24} $	$ {\bf{[46,45,2]}} $
$ C_{25} $	$ [48,38,4] $
$ C_{26} $	$ [48,39,2] $
$ C_{27} $	$ [48,39,2] $
$ C_{28} $	$ [48,40,4] $
$ C_{29} $	$ [50,35,4] $
$ C_{30} $	$ [50,36,4] $
$ C_{31} $	$ [50,39,4] $
$ C_{32} $	$ [50,40,3] $
$ C_{33} $	$ [52,38,4] $
$ C_{34} $	$ [52,40,5] $
$ C_{35} $	$ [54,46,2] $
$ C_{36} $	$ [54,47,2] $
$ C_{37} $	$ [54,48,2] $
$ C_{38} $	$ [54,49,2] $
$ C_{39} $	$ [54,50,2] $
$ C_{40} $	$ {\bf{[54,51,2]}} $
$ C_{41} $	$ {\bf{[54,52,2]}} $
$ C_{42} $	$ {\bf{[54,53,2]}} $
$ C_{43} $	$ [56,42,2] $
$ C_{44} $	$ [56,44,4] $
$ C_{45} $	$ [56,45,4] $
$ C_{46} $	$ [56,46,4] $
$ C_{47} $	$ [58,43,6] $
$ C_{48} $	$ [58,44,6] $

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