# American Institute of Mathematical Sciences

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

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.

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] 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. [2] L. Adleman, Molecular computation of the solutions to combinatorial problems, Science, 266 (1994), 1021-1024.  doi: 10.1126/science.7973651. [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. [4] D. Boneh, C. Dunworth and R. Lipton, Breaking DES using molecular computer, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 27 (1996), 37-65. [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. [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. [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. [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. [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. [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. [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. [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. [13] T. Hurley, Group rings and rings of matrices, Int. Jour. Pure and Appl. Math, 31 (2006), 319-335. [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. [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. [16] C. P. Milies and S. K. Sehgal, An Introduction to Group Rings, , Kluwer Academic Publishers, Dordrecht, 2002. [17] S. K. Sehgal, Units in Integral Group Rings, Pitman Monographs and Surveys in Pure and Applied Mathematics, 69. Longman Scientific & Technical, Harlow, 1993.

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. [2] L. Adleman, Molecular computation of the solutions to combinatorial problems, Science, 266 (1994), 1021-1024.  doi: 10.1126/science.7973651. [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. [4] D. Boneh, C. Dunworth and R. Lipton, Breaking DES using molecular computer, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 27 (1996), 37-65. [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. [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. [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. [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. [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. [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. [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. [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. [13] T. Hurley, Group rings and rings of matrices, Int. Jour. Pure and Appl. Math, 31 (2006), 319-335. [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. [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. [16] C. P. Milies and S. K. Sehgal, An Introduction to Group Rings, , Kluwer Academic Publishers, Dordrecht, 2002. [17] S. K. Sehgal, Units in Integral Group Rings, Pitman Monographs and Surveys in Pure and Applied Mathematics, 69. Longman Scientific & Technical, Harlow, 1993.
Reversible Group Codes from ${\mathcal{G}}_1$ over ${\mathbb{F}}_4$
 $C_i$ $[n,k,d]$ $C_1$ $[24,11,6]$ $C_2$ $[24,10,8]$ $C_3$ $[26,18,4]$ $C_4$ $[28,16,4]$ $C_5$ $[28,17,4]$ $C_6$ $[28,18,4]$ $C_7$ $[28,19,4]$ $C_8$ $[28,20,4]$ $C_9$ $[30,18,6]$ $C_{10}$ $[30,19,6]$ $C_{11}$ ${\bf{[30,20,6]}}$ $C_{12}$ $[32,18,4]$ $C_{13}$ $[32,20,4]$ $C_{14}$ $[32,22,4]$
 $C_i$ $[n,k,d]$ $C_1$ $[24,11,6]$ $C_2$ $[24,10,8]$ $C_3$ $[26,18,4]$ $C_4$ $[28,16,4]$ $C_5$ $[28,17,4]$ $C_6$ $[28,18,4]$ $C_7$ $[28,19,4]$ $C_8$ $[28,20,4]$ $C_9$ $[30,18,6]$ $C_{10}$ $[30,19,6]$ $C_{11}$ ${\bf{[30,20,6]}}$ $C_{12}$ $[32,18,4]$ $C_{13}$ $[32,20,4]$ $C_{14}$ $[32,22,4]$
Reversible Group Codes from ${\mathcal{G}}_2$ over ${\mathbb{F}}_4$
 $C_i$ $[n,k,d]$ $C_{15}$ $[40,24,5]$ $C_{16}$ $[40,28,4]$ $C_{17}$ $[40,28,5]$ $C_{18}$ $[40,28,6]$ $C_{19}$ $[40,29,4]$ $C_{20}$ $[40,30,5]$ $C_{21}$ $[44,34,2]$ $C_{22}$ $[44,32,4]$ $C_{23}$ ${\bf{[46,44,2]}}$ $C_{24}$ ${\bf{[46,45,2]}}$ $C_{25}$ $[48,38,4]$ $C_{26}$ $[48,39,2]$ $C_{27}$ $[48,39,2]$ $C_{28}$ $[48,40,4]$ $C_{29}$ $[50,35,4]$ $C_{30}$ $[50,36,4]$ $C_{31}$ $[50,39,4]$ $C_{32}$ $[50,40,3]$ $C_{33}$ $[52,38,4]$ $C_{34}$ $[52,40,5]$ $C_{35}$ $[54,46,2]$ $C_{36}$ $[54,47,2]$ $C_{37}$ $[54,48,2]$ $C_{38}$ $[54,49,2]$ $C_{39}$ $[54,50,2]$ $C_{40}$ ${\bf{[54,51,2]}}$ $C_{41}$ ${\bf{[54,52,2]}}$ $C_{42}$ ${\bf{[54,53,2]}}$ $C_{43}$ $[56,42,2]$ $C_{44}$ $[56,44,4]$ $C_{45}$ $[56,45,4]$ $C_{46}$ $[56,46,4]$ $C_{47}$ $[58,43,6]$ $C_{48}$ $[58,44,6]$
 $C_i$ $[n,k,d]$ $C_{15}$ $[40,24,5]$ $C_{16}$ $[40,28,4]$ $C_{17}$ $[40,28,5]$ $C_{18}$ $[40,28,6]$ $C_{19}$ $[40,29,4]$ $C_{20}$ $[40,30,5]$ $C_{21}$ $[44,34,2]$ $C_{22}$ $[44,32,4]$ $C_{23}$ ${\bf{[46,44,2]}}$ $C_{24}$ ${\bf{[46,45,2]}}$ $C_{25}$ $[48,38,4]$ $C_{26}$ $[48,39,2]$ $C_{27}$ $[48,39,2]$ $C_{28}$ $[48,40,4]$ $C_{29}$ $[50,35,4]$ $C_{30}$ $[50,36,4]$ $C_{31}$ $[50,39,4]$ $C_{32}$ $[50,40,3]$ $C_{33}$ $[52,38,4]$ $C_{34}$ $[52,40,5]$ $C_{35}$ $[54,46,2]$ $C_{36}$ $[54,47,2]$ $C_{37}$ $[54,48,2]$ $C_{38}$ $[54,49,2]$ $C_{39}$ $[54,50,2]$ $C_{40}$ ${\bf{[54,51,2]}}$ $C_{41}$ ${\bf{[54,52,2]}}$ $C_{42}$ ${\bf{[54,53,2]}}$ $C_{43}$ $[56,42,2]$ $C_{44}$ $[56,44,4]$ $C_{45}$ $[56,45,4]$ $C_{46}$ $[56,46,4]$ $C_{47}$ $[58,43,6]$ $C_{48}$ $[58,44,6]$
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