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doi: 10.3934/amc.2021056
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Reversible $ G $-codes over the ring $ {\mathcal{F}}_{j,k} $ with applications to DNA codes

Yasemin Cengellenmis 1, , Abdullah Dertli 2, , Steven T. Dougherty 3, , Adrian Korban 4,, , Serap Şahinkaya 5, and  Deniz Ustun 6,

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

Received  March 2021 Revised  September 2021 Early access December 2021

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: Group Codes, Linear Codes, Codes over Rings, Gray Maps, DNA Codes.
Mathematics Subject Classification: Primary: 94B05; Secondary: 16S34.
Citation: Yasemin Cengellenmis, Abdullah Dertli, Steven T. Dougherty, Adrian Korban, Serap Şahinkaya, Deniz Ustun. Reversible $ G $-codes over the ring $ {\mathcal{F}}_{j,k} $ with applications to DNA codes. Advances in Mathematics of Communications, doi: 10.3934/amc.2021056
References:
[1]

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Y. Cengellenmis, A. Dertli and S. T. Dougherty, Codes over an infinite family of rings with a Gray map, Des. Codes Cryptogr., 72 (2014), 559-580.  doi: 10.1007/s10623-012-9787-y.  Google Scholar

[7]

Y. Cengellenmis, A. Dertli, S. T. Dougherty, A. Korban, S. Sahinkaya and D. Ustun, Generator matrices for the manuscript entitled reversible $G$–Codes over the ring ${\mathcal{F}}_{j, k}$ with applications to DNA codes, available at https://sites.google.com/view/adriankorban/generator-matrices. Google Scholar

[8]

S. T. Dougherty, Algebraic Coding Theory Over Finite Commutative Rings, Springer Briefs in Mathematics, Springer, 2017. doi: 10.1007/978-3-319-59806-2.  Google Scholar

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S. T. Dougherty, J. Gildea, R. Taylor and A. Tylshchak, Group rings, $G$-codes and constructions of self-dual and formally self-dual codes, Des. Codes Cryptogr., 86 (2018), 2115-2138.  doi: 10.1007/s10623-017-0440-7.  Google Scholar

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S. T. Dougherty, B. Yildiz and S. Karadeniz, Codes over $R_k$, Gray maps and their binary images, Finite Fields Appl., 17 (2011), 205-219.  doi: 10.1016/j.ffa.2010.11.002.  Google Scholar

[11]

S. T. Dougherty, B. Yildiz and S. Karadeniz, Self-dual codes over $R_k$ and binary self-dual codes, Eur. J. Pure Appl. Math., 6 (2013), 89-106.   Google Scholar

[12]

F. Gursoy, E. S. Oztas and A. Ozkan, Reversible DNA codes over a family of non-chain rings, https://arXiv.org/pdf/1711.02385.pdf. Google Scholar

[13]

T. Hurley, Group rings and rings of matrices, Int. Jour. Pure and Appl. Math, 31 (2006), 319-335.   Google Scholar

[14]

A. Korban, S. Sahinkaya and D. Ustun, An Application of the virus optimization algorithm to the problem of finding extremal binary self-dual codes, arXiv: 2103.07739v1. Google Scholar

[15]

M. Mansuripur, P. K. Khulbe, S. M. Kubler, J. W. Perry, M. S. Giridhar and N. Peyghambarian, Information storage and retrieval using macromolecules as storage media, Ptical Data Storage, OSA Technical Digest Series (Optical Society of America), paper TuC2, 2003. Google Scholar

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C. P. Milies and S. K. Sehgal, An Introduction to Group Rings, , Kluwer Academic Publishers, Dordrecht, 2002.  Google Scholar

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S. K. Sehgal, Units in Integral Group Rings, Pitman Monographs and Surveys in Pure and Applied Mathematics, 69. Longman Scientific & Technical, Harlow, 1993.  Google Scholar

show all references

References:
[1]

T. Abualrub, A. Ghrayeb and X. N. Zeng, Construction of cyclic codes over $GF(4)$ for DNA computing, J. Franklin Inst., 343 (2006), 448-457.  doi: 10.1016/j.jfranklin.2006.02.009.  Google Scholar

[2]

L. Adleman, Molecular computation of the solutions to combinatorial problems, Science, 266 (1994), 1021-1024.  doi: 10.1126/science.7973651.  Google Scholar

[3]

L. Adleman, P. W. K. Rothemund, S. Rowies and E. Winfree, On applying molecular computation to the data encryption standard, J. Comp. Biology, 6 (1999), 53-63.  doi: 10.1089/cmb.1999.6.53.  Google Scholar

[4]

D. Boneh, C. Dunworth and R. Lipton, Breaking DES using molecular computer, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 27 (1996), 37-65.   Google Scholar

[5]

W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235-265.  doi: 10.1006/jsco.1996.0125.  Google Scholar

[6]

Y. Cengellenmis, A. Dertli and S. T. Dougherty, Codes over an infinite family of rings with a Gray map, Des. Codes Cryptogr., 72 (2014), 559-580.  doi: 10.1007/s10623-012-9787-y.  Google Scholar

[7]

Y. Cengellenmis, A. Dertli, S. T. Dougherty, A. Korban, S. Sahinkaya and D. Ustun, Generator matrices for the manuscript entitled reversible $G$–Codes over the ring ${\mathcal{F}}_{j, k}$ with applications to DNA codes, available at https://sites.google.com/view/adriankorban/generator-matrices. Google Scholar

[8]

S. T. Dougherty, Algebraic Coding Theory Over Finite Commutative Rings, Springer Briefs in Mathematics, Springer, 2017. doi: 10.1007/978-3-319-59806-2.  Google Scholar

[9]

S. T. Dougherty, J. Gildea, R. Taylor and A. Tylshchak, Group rings, $G$-codes and constructions of self-dual and formally self-dual codes, Des. Codes Cryptogr., 86 (2018), 2115-2138.  doi: 10.1007/s10623-017-0440-7.  Google Scholar

[10]

S. T. Dougherty, B. Yildiz and S. Karadeniz, Codes over $R_k$, Gray maps and their binary images, Finite Fields Appl., 17 (2011), 205-219.  doi: 10.1016/j.ffa.2010.11.002.  Google Scholar

[11]

S. T. Dougherty, B. Yildiz and S. Karadeniz, Self-dual codes over $R_k$ and binary self-dual codes, Eur. J. Pure Appl. Math., 6 (2013), 89-106.   Google Scholar

[12]

F. Gursoy, E. S. Oztas and A. Ozkan, Reversible DNA codes over a family of non-chain rings, https://arXiv.org/pdf/1711.02385.pdf. Google Scholar

[13]

T. Hurley, Group rings and rings of matrices, Int. Jour. Pure and Appl. Math, 31 (2006), 319-335.   Google Scholar

[14]

A. Korban, S. Sahinkaya and D. Ustun, An Application of the virus optimization algorithm to the problem of finding extremal binary self-dual codes, arXiv: 2103.07739v1. Google Scholar

[15]

M. Mansuripur, P. K. Khulbe, S. M. Kubler, J. W. Perry, M. S. Giridhar and N. Peyghambarian, Information storage and retrieval using macromolecules as storage media, Ptical Data Storage, OSA Technical Digest Series (Optical Society of America), paper TuC2, 2003. Google Scholar

[16]

C. P. Milies and S. K. Sehgal, An Introduction to Group Rings, , Kluwer Academic Publishers, Dordrecht, 2002.  Google Scholar

[17]

S. K. Sehgal, Units in Integral Group Rings, Pitman Monographs and Surveys in Pure and Applied Mathematics, 69. Longman Scientific & Technical, Harlow, 1993.  Google Scholar

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] $
Table Options

Download as excel

$ 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] $
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] $
Table Options

Download as excel

$ 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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