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Designs, Codes and Cryptography

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12-10-2023

Minimal and optimal binary codes obtained using \(C_D\)-construction over the non-unital ring I

Authors: Vidya Sagar, Ritumoni Sarma

Published in: Designs, Codes and Cryptography | Issue 1/2024

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Abstract

In this article, we construct linear codes over the commutative non-unital ring I of size four. We obtain their Lee-weight distributions and study their binary Gray images. Under certain mild conditions, these classes of binary codes are minimal and self-orthogonal. All codes in this article are few-weight codes. Besides, an infinite class of these binary codes consists of distance optimal codes with respect to the Griesmer bound.

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Alahmadi A., Alkathiry A., Altassan A., Bonnecaze A., Shoaib H., Solé P.: The build-up construction of quasi self-dual codes over a non-unital ring. J. Algebra Appl. (2020). https://doi.org/10.1142/S0219498822501432.CrossRef

Alahmadi A., Alkathiry A., Altassan A., Bonnecaze A., Shoaib H., Solé P.: The build-up construction over a commutative non-unital ring. Des. Codes Cryptogr. 90(12), 3003–3010 (2022).MathSciNetCrossRef

Alahmadi A., Altassan A., Basaffar W., Bonnecaze A., Shoaib H., Solé P.: Quasi Type IV codes over a non-unital ring. Appl. Algebra Eng. Comm. Comput. 32, 217–228 (2021).MathSciNetCrossRef

Alahmadi A., Altassan A., Basaffar W., Shoaib H., Bonnecaze A., Solé P.: Type IV codes over a non-unital ring. J. Algebra Appl. 21(07), 2250142 (2022).MathSciNetCrossRef

Ashikhmin A., Barg A.: Minimal vectors in linear codes. IEEE Trans. Inform. Theory 44(5), 2010–2017 (1998).MathSciNetCrossRef

Ashikhmin A., Knill E.: Nonbinary quantum stabilizer codes. IEEE Trans. Inf. Theory 47(7), 3065–3072 (2001).MathSciNetCrossRef

Bonisoli A.: Every equidistant linear code is a sequence of dual Hamming codes. Ars Combinatoria 18, 181–186 (1984).MathSciNet

Calderbank R., Kantor W.M.: The geometry of two-weight codes. Bull. London Math. Soc. 18(2), 97–122 (1986).MathSciNetCrossRef

Calderbank A.R., Rains E.M., Shor P.W., Sloane N.J.A.: Quantum error correction via codes over GF (4). IEEE Trans. Inform. Theory 44(4), 1369–1387 (1998).MathSciNetCrossRef

10.

Chang S., Hyun J.Y.: Linear codes from simplicial complexes. Des. Codes Cryptogr. 86(10), 2167–2181 (2018).MathSciNetCrossRef

11.

Delsarte P.: Weights of linear codes and strongly regular normed spaces. Discret. Math. 3(1–3), 47–64 (1972).MathSciNetCrossRef

12.

Ding C., Helleseth T., Kløve T., Wang X.: A generic construction of Cartesian authentication codes. IEEE Trans. Inform. Theory 53(6), 2229–2235 (2007).MathSciNetCrossRef

13.

Ding K., Ding C.: A class of two-weight and three-weight codes and their applications in secret sharing. IEEE Trans. Inform. Theory 61(11), 5835–5842 (2015).MathSciNetCrossRef

14.

Dougherty S.T., Gaborit P., Harada M., Munemasa A., Solé P.: Type IV self-dual codes over rings. IEEE Trans. Inform. Theory 45(7), 2345–2360 (1999).MathSciNetCrossRef

15.

Dougherty S.T., Gaborit P., Harada M., Solé P.: Type II Codes Over \({\mathbb{F} }_2 + u{\mathbb{F} }_2\). IEEE Trans. Inform. Theory 45(1), 32–45 (1999).MathSciNetCrossRef

16.

Du Z., Li C., Mesnager S.: Constructions of self-orthogonal codes from hulls of BCH codes and their parameters. IEEE Trans. Inform. Theory 66(11), 6774–6785 (2020).MathSciNetCrossRef

17.

Fine B.: Classification of finite rings of order \(p^2\). Math. Mag. 66(4), 248–252 (1993).MathSciNetCrossRef

18.

Gan C., Li C., Mesnager S., Qian H.: On hulls of some primitive BCH codes and self-orthogonal codes. IEEE Trans. Inform. Theory 67(10), 6442–6455 (2021).MathSciNetCrossRef

19.

Grassl, M.: Bounds on the minimum distance of linear codes http://www.codetables.de, Accessed on 1, 2023

20.

Griesmer J.H.: A bound for error correcting codes. IBM J. Res. Dev. 4(5), 532–542 (1960).MathSciNetCrossRef

21.

Hammons A.R., Kumar P.V., Calderbank A.R., Sloane N.J., Solé P.: The \({\mathbb{Z} }_4\)-linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inform. Theory 40(2), 301–319 (1994).MathSciNetCrossRef

22.

Hyun J.Y., Kim H.K., Wu Y., Yue Q.: Optimal minimal linear codes from posets. Des. Codes Cryptogr. 88(12), 2475–2492 (2020).MathSciNetCrossRef

23.

Hyun J.Y., Lee J., Lee Y.: Infinite families of optimal linear codes constructed from simplicial complexes. IEEE Trans. Inform. Theory 66(11), 6762–6775 (2020).MathSciNetCrossRef

24.

Huffman W.C., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003).CrossRef

25.

Kim J.L., Ohk D.E.: DNA codes over two noncommutative rings of order four. J. Appl. Math. Comput. 68(3), 2015–2038 (2022).MathSciNetCrossRef

26.

Kim J.L., Roe Y.G.: Construction of quasi self-dual codes over a commutative non-unital ring of order 4. Appl. Algebra Eng. Comm. Comput. (2022). https://doi.org/10.1007/s00200-022-00553-8.CrossRef

27.

Kløve T.: Codes for Error Detection. World Scientific, Singapore (2007).CrossRef

28.

Liu H., Yu Z.: Linear Codes from Simplicial Complexes over \({\mathbb{F} } _ {2^ n} \). (2023). arXiv preprint arXiv:2303.09292.

29.

MacWilliams F.J., Sloane J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977).

30.

Massey J.L.: Minimal codewords and secret sharing. In: Proceedings of the 6th Joint Swedish-Russian International Workshop on Information Theory, pp. 276–279 (1993)

31.

Sagar V., Sarma R.: Octanary linear codes using simplicial complexes. Cryptogr. Commun. 15(3), 599–616 (2023). https://doi.org/10.1007/s12095-022-00617-z.

32.

Sagar V., Sarma R.: Certain binary minimal codes constructed using simplicial complexes. (2023). arXiv preprint arXiv:2211.15747v2.

33.

Sagar V., Sarma R.: Codes over the non-unital non-commutative ring \(E\) using simplicial complexes. (2023). arXiv preprint arXiv:2304.06758.

34.

Shi M., Li X.: A new family of optimal binary few-weight codes from simplicial complexes. IEEE Commun. Lett. 25(4), 1048–1051 (2021).CrossRef

35.

Shi M., Li X.: Few-weight codes over a non-chain ring associated with simplicial complexes and their distance optimal Gray image. Finite Fields Appl. 80, 101994 (2022).MathSciNetCrossRef

36.

Shi M., Li X.: New classes of binary few weight codes from trace codes over a chain ring. J. Appl. Math. Comput. 68(3), 1869–1880 (2022).MathSciNetCrossRef

37.

Shi M., Guan Y., Wang C., Solé P.: Few-weight codes from trace codes over \(R_k\). Bull. Aust. Math. Soc. 98(1), 167–174 (2018).MathSciNetCrossRef

38.

Shi M., Qian L., Helleseth T., Solé P.: Five-weight codes from three-valued correlation of M-sequences. Adv. Math. Commun. (2021). https://doi.org/10.3934/amc.2021022.CrossRef

39.

Shi M., Wang S., Kim J.L., Solé P.: Self-orthogonal codes over a non-unital ring and combinatorial matrices. Des. Codes Cryptogr. (2021). https://doi.org/10.1007/s10623-021-00948-7.CrossRef

40.

Shi M., Xuan W., Solé P.: Two families of two-weight codes over \({\mathbb{F} }_4\). Des. Codes Cryptogr. 88(12), 2493–2505 (2020).MathSciNetCrossRef

41.

Shi M., Li S., Kim J.L., Solé P.: LCD and ACD codes over a noncommutative non-unital ring with four elements. Cryptogr. Commun. 14(3), 627–640 (2022). https://doi.org/10.1007/s12095-021-00545-4.

42.

Wang S., Shi M.: Few-weight \({\varvec {{{\mathbb{Z} }}}} _p \varvec {{{\mathbb{Z} }}} _p [u]\)-additive codes from down-sets. J. Appl. Math. Comput. 68(4), 2381–2388 (2022).MathSciNetCrossRef

43.

Wu Y., Lee Y.: Binary LCD codes and self-orthogonal codes via simplicial complexes. IEEE Commun. Lett. 24(6), 1159–1162 (2020).CrossRef

44.

Wu Y., Lee Y.: Self-orthogonal codes constructed from posets and their applications in quantum communication. Mathematics 8(9), 1495 (2020).CrossRef

45.

Wu Y., Li C., Xiao F.: Quaternary linear codes and related binary subfield codes. IEEE Trans. Inform. Theory 68(5), 3070–3080 (2022).MathSciNetCrossRef

46.

Wu Y., Zhu X., Yue Q.: Optimal few-weight codes from simplicial complexes. IEEE Trans. Inform. Theory 66(6), 3657–3663 (2020).MathSciNetCrossRef

47.

Yuan J., Ding C.: Secret sharing schemes from three classes of linear codes. IEEE Trans. Inform. Theory 52(1), 206–212 (2006).MathSciNetCrossRef

48.

Zhou Z., Li X., Tang C., Ding C.: Binary LCD codes and self-orthogonal codes from a generic construction. IEEE Trans. Inform. Theory 65(1), 16–27 (2018).MathSciNetCrossRef

49.

Zhu X., Wei Y.: Few-weight quaternary codes via simplicial complexes. AIMS Math. 6(5), 5124–5132 (2021).MathSciNetCrossRef

Title: Minimal and optimal binary codes obtained using -construction over the non-unital ring I
Authors: Vidya Sagar
Ritumoni Sarma
Publication date: 12-10-2023
Publisher: Springer US
Published in: Designs, Codes and Cryptography / Issue 1/2024
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-023-01299-1

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Minimal and optimal binary codes obtained using \(C_D\)-construction over the non-unital ring I

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