One Weight Codes Over the Ring $F_q[v]/(v^s-1)$

Year 2023, Volume: 8 Issue: 1, 35 - 47, 30.04.2023

Mustafa Sarı

https://doi.org/10.30931/jetas.1152408

Abstract

In this study, we obtain one-Lee weight codes over a class of nonchain rings and study their structures.
We give an explicit construction for one-Lee weight codes. A method to derive more one-Lee weight codes from given a one-Lee weight code is also represented. By defining and making use of a distance-preserving Gray map, we get a family of optimal one-Hamming weight codes over finite fields.

Keywords

Linear codes , one-Lee weight codes , gray map , optimal codes

References

• [1] Agrell, E., Vardy, A., Zeger, K., "Upper bounds for constant-weight codes", IEEE Transactions on Information Theory 46 (2000) : 2373-2395.
• [2] Bonisoli, A., "Every equidistant linear code is a sequence of dual Hamming codes", Ars Combinatoria 18 (1983) : 181-186.
• [3] Brouwer, A.E., Shearer, J.B., Sloane, N.J.A., Smith, W.D., "A new table of constant-weight codes", IEEE Transactions on Information Theory 36 (1990) : 1334-1380.
• [4] Carlet, C., "One-weight -linear codes", Coding, Cryptography and Related Areas (2000) : 57-72.
• [5] Kuekes, P.J., Robinett, W., Roth, R.M., Seroussi, G., Snider, G.S., Williams, R.S., "Resistor-logic demultiplexers for nano electronics based on constant-weight codes", Nanotechnology 17 (2006) : 1052-1061.
• [6] Li, T., Shi, M., Lin B., Wu, W., "One and two-weight additive codes", Chinese Journal of Electronics 30(1) (2021) : 72-76.
• [7] Li, S., Shi, M. "Two infinite families of two-weight codes over ", Journal of Applied Mathematics and Computing 69 (2022) : 201-218.
• [8] Lint, J.V., Tolhuizenon, L., "On perfect ternary constant-weight codes", Design Codes and Cryptography 18 (1999) : 231-234.
• [9] Moon, J.N.J., Hughes, L.A., Smith, D.H., "Assignment of frequency lists in frequency hopping networks", IEEE Transactions on Vehicular Technology 54(3) (2005) : 1147-1159.
• [10] Peterson, W.W., Weldon Jr, E.J., "Error-correcting codes", The MIT Press, 1972.
• [11] Rains, E.M., Sloane, N.J.A., "Table of constant-weight binary codes", [Online]. Available: http://www.research.att.com/~njas/codes/Andw/.
• [12] Sari, M., Siap, I., Siap, V., "One-Homogeneous weight codes over finite chain rings", Bulletin of the Korean Mathematical Society 52(6) (2015) : 2011-2023.
• [13] Shi, M., "Optimal -ary codes from one-weight linear codes over ", Chinese Journal of Electronics 22(4) (2013) : 799-802.
• [14] Shi, M., Zhu, S., Yang, S., "A class of optimal -ary codes from one-weight codes over ", Journal of the Franklin Institute 350 (2013) : 929-937.
• [15] Shi, M., Wang, Y., "Optimal binary codes from one-Lee weight codes and two-Lee weight projective codes over ", Journal of Systems Science and Complexity 27(4) (2014) : 795-810.
• [16] Shi, M., Xu, L., Yang, G., "A Note on One Weight and Two Weight Projective -codes", IEEE Transactions on Information Theory 63(1) (2017) : 177-182.
• [17] Shi, M., Wang, C., Wu, R., Hu, Y., Chang, Y.,"One-weight and two-weight -additive codes", Cryptography and Communications 12 (2020) : 443-454.
• [18] Smith, D.M., Montemanni, R., "Bounds for constant- weight binary codes with ", [Online]. Available: http://www.idsia.ch/~roberto/Andw29/.
• [19] Wood, J.A., "The structure of linear codes of constant weight", Transactions of the American Mathematical Society 35 (2002) : 1007-1026.