A class of constacyclic codes containing formally self-dual and isodual codes

Yıl 2020, Volume: 7 Issue: 1 (Special Issue in Algebraic Coding Theory: New Trends and Its Connections), 21 - 33, 29.02.2020

Manjit Singh

https://doi.org/10.13069/jacodesmath.645018

Öz

In this paper, we investigate a class of constacyclic codes which contains isodual codes and formally self-dual codes. Further, we introduce a recursive approach to obtain the explicit factorization of $x^{2^m\ell^n}-\mu_k\in\mathbb{F}_q[x]$, where $n, m$ are positive integers and $\mu_k$ is an element of order $\ell^k$ in $\mathbb{F}_q$. Moreover, we give many examples of interesting isodual and formally self-dual constacyclic codes.

Anahtar Kelimeler

Constacyclic codes , Weight distributions , Isodual codes , Formally self-dual codes

Kaynakça

• [1] N. Aydin, I. Siap, D. K. Ray–Chaudhuri, The structure of 1–generator quasi–twisted codes and new linear codes, Des. Codes Cryptogr. 24(3) (2001) 313–326.
• [2] C. Bachoc, T. A. Gulliver, M. Harada, Isodual codes over $\mathbb{Z}_{2k}$ and isodual lattices, J. Algebra Combin. 12(3) (2000) 223–240.
• [3] G. K. Bakshi, M. Raka, A class of constacyclic codes over a finite field, Finite Field Appl. 18(2) (2012) 362–377.
• [4] T. Blackford, Negacyclic duadic codes, Finite Fields Appl. 14(4) (2008) 930–943.
• [5] T. Blackford, Isodual constacyclic codes, Finite Fields Appl. 24 (2013) 29–44.
• [6] B. Chen, Y. Fan, L. Lin, H. Liu, Constacyclic codes over finite fields, Finite Fields Appl. 18(6) (2012) 1217–1231.
• [7] H. Q. Dinh, C. Li, Q. Yue, Recent progress on weight distributions of cyclic codes over finite fields, J. Algebra Comb. Discrete Struct. Appl. 2(1) (2015) 39–63.
• [8] H. Q. Dinh, Repeated–root constacyclic codes of length $2p^s$, Finite Fields Appl. 18(1) (2012) 133–143.
• [9] W. C. Huffman, V. Pless, Fundamentals of Error–Correcting Codes, Cambridge University Press, 2003.
• [10] G. T. Kennedy, V. Pless, On designs and formally self–dual codes, Des. Codes Cryptogr. 4(1) (1994) 43–55.
• [11] F. Li, Q. Yue, The primitive idempotents and weight distributions of irreducible constacyclic codes, Des. Codes Cryptogr. 86(4) (2018) 771–784.
• [12] R. Lidl, H. Niederreiter, Introduction to Finite Fields and Their Applications, Cambridge University Press, 1986.
• [13] S. Ling, C. Xing, Coding Theory: A First Course, Cambridge University Press, 2004.
• [14] J. L. Massey, Minimal codewords and secret sharing, Proc. 6th Joint Swedish–Russian Workshop on Information Theory, Mölle, Sweden, (1993) 276–279.
• [15] M. Singh, Some subgroups of $\mathbb{F}_q^*$ and explicit factors of $x^{2^nd}-1\in\mathbb{F}_q[x]$ Transactions on Combinatorics (2019) doi: 10.22108/TOC.2019.114742.1612.
• [16] M. Singh, S. Batra, Some special cyclic codes of length $2^n$, J. Algebra Appl. 16(1) (2017) 17 pages.
• [17] M. Singh, S. Batra, Weight distribution of a class of cyclic codes of length $2^n$, J. Algebra Comb. Discrete Appl. 6(1) (2018) 1–11.
• [18] X. Zhu, Q. Yue, L. Hu, Weight distributions of cyclic codes of length $l^m$, Finite Fields Appl. 31 (2015) 241–257.