A Class of Constacyclic Codes Containing Formally Self-dual and Isodual Codes
Keywords:
Constacyclic codes, Weight distributions, Isodual codes, Formally self-dual codes
Abstract
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.
References
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C. Bachoc, T. A. Gulliver, M. Harada, Isodual codes overZ2kand isodual lattices, J. AlgebraCombin. 12(3) (2000) 223–240.
G. K. Bakshi, M. Raka, A class of constacyclic codes over a finite field, Finite Field Appl. 18(2)(2012) 362–377.
T. Blackford, Negacyclic duadic codes, Finite Fields Appl. 14(4) (2008) 930–943.
T. Blackford, Isodual constacyclic codes, Finite Fields Appl. 24 (2013) 29–44.
B. Chen, Y. Fan, L. Lin, H. Liu, Constacyclic codes over finite fields, Finite Fields Appl. 18(6) (2012)1217–1231.
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.
H. Q. Dinh, Repeated–root constacyclic codes of length2ps, Finite Fields Appl. 18(1) (2012) 133–143.
W. C. Huffman, V. Pless, Fundamentals of Error–Correcting Codes, Cambridge University Press,2003.
G. T. Kennedy, V. Pless, On designs and formally self–dual codes, Des. Codes Cryptogr. 4(1) (1994)43–55.
F. Li, Q. Yue, The primitive idempotents and weight distributions of irreducible constacyclic codes,Des. Codes Cryptogr. 86(4) (2018) 771–784.
R. Lidl, H. Niederreiter, Introduction to Finite Fields and Their Applications, Cambridge UniversityPress, 1986.
S. Ling, C. Xing, Coding Theory: A First Course, Cambridge University Press, 2004.
J. L. Massey, Minimal codewords and secret sharing, Proc. 6th Joint Swedish–Russian Workshop onInformation Theory, Mölle, Sweden, (1993) 276–279.
M. Singh, Some subgroups ofF∗qand explicit factors ofx2nd−1∈Fq[x], Transactions on Combina-torics (2019) doi: 10.22108/TOC.2019.114742.1612.
M. Singh, S. Batra, Some special cyclic codes of length2n, J. Algebra Appl. 16(1) (2017) 17 pages.
M. Singh, S. Batra, Weight distribution of a class of cyclic codes of length2n, J. Algebra Comb.Discrete Appl. 6(1) (2018) 1–11.
X. Zhu, Q. Yue, L. Hu, Weight distributions of cyclic codes of lengthlm, Finite Fields Appl. 31(2015) 241–257.33
Published
2020-01-15
