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Designs, Codes and Cryptography

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10-01-2023

A general family of Plotkin-optimal two-weight codes over \(\mathbb {Z}_4\)

Authors: Hopein Christofen Tang, Djoko Suprijanto

Published in: Designs, Codes and Cryptography | Issue 5/2023

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Abstract

We obtained all possible parameters of Plotkin-optimal two-Lee weight projective codes over \(\mathbb {Z}_4,\) together with their weight distributions. We show the existence of codes with these parameters as well as their weight distributions by constructing an infinite family of two-weight codes. Previously known codes constructed by Shi et al. (Des Codes Cryptogr 88(12): 2493–2505, 2020) can be derived as a special case of our results. We also prove that the Gray image of any Plotkin-optimal two-Lee weight projective codes over \(\mathbb {Z}_4\) has the same parameters and weight distribution as some two-weight binary projective codes of type SU1 in the sense of Calderbank and Kantor (Bull Lond Math Soc 18: 97–122, 1986).

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Calderbank R.: On uniformly packed \([n, n-k,4]\) codes over \(GF(q)\) and a class of caps in \(PG(k-1, q),\). J. Lond. Math. Soc. 26(2), 365–384 (1982).MathSciNetCrossRefMATH

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

Carlet C.: One-weight \({\mathbb{Z} }\)-linear codes. In: Buchmann J., et al. (eds.) Coding Theory, Cryptography and Related Areas. Springer, New York (2000).

Delsarte P.: Weights of linear codes and strongly regular normed spaces. Discret Math. 3, 47–62 (1972).MathSciNetCrossRefMATH

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

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

Kløve T.: Support weight distribution of linear codes. Discret Math. 106–107, 311–316 (1992).MathSciNetCrossRefMATH

Li S., Shi M.: Two infinite families of two-weight codes over \({\mathbb{Z} }_{2^m}\). J. Appl. Math. Comput. (2022). https://doi.org/10.1007/s12190-022-01736-9.CrossRef

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

10.

Shi M., Guan Y., Solé P.: Two new families of two-weight codes. IEEE Trans. Inf. Theory 63(10), 6240–6246 (2017).MathSciNetCrossRefMATH

11.

Shi M., Honold T., Solé P., Qiu Y., Wu R., Sepasdar Z.: The geometry of two-weight codes over \({\mathbb{Z} }_{p^m}\). IEEE Trans. Inf. Theory 67(12), 7769–7781 (2021).CrossRefMATH

12.

Shi M., Sepasdar Z., Alahmadi A., Solé P.: On two-weight \({\mathbb{Z} }_{2^k}\)-codes. Des. Codes Cryptogr. 86(6), 1201–1209 (2018).MathSciNetCrossRefMATH

13.

Shi M., Wang Y.: Optimal binary codes from one-Lee weight codes and two-Lee weight projective codes over \({\mathbb{Z} }_4,\). J. Syst. Sci. Complex. 27(4), 795–810 (2014).MathSciNetCrossRefMATH

14.

Shi M., Xu L.L., Yang G.: A note on one weight and two weight projective \({\mathbb{Z} }_4\)-codes. IEEE Trans. Inf. Theory 63(1), 177–182 (2017).CrossRef

15.

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

16.

Tang H.C., Suprijanto D.: New optimal linear codes over \({\mathbb{Z} }_4\). Bull. Aust. Math. Soc. (2022). https://doi.org/10.1017/S0004972722000399.CrossRef

17.

Wan Z.: Quaternary Codes. World Scientifc, Singapore (1997).CrossRefMATH

18.

Wyner A.D., Graham R.L.: An upper bound on minimum distance for a \(k\)-ary code. Inf. Control 13(1), 46–52 (1968).MathSciNetCrossRefMATH

Title: A general family of Plotkin-optimal two-weight codes over
Authors: Hopein Christofen Tang
Djoko Suprijanto
Publication date: 10-01-2023
Publisher: Springer US
Published in: Designs, Codes and Cryptography / Issue 5/2023
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-022-01176-3

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A general family of Plotkin-optimal two-weight codes over \(\mathbb {Z}_4\)

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