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

Yıl 2020, , 21 - 33, 29.02.2020
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.

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.
Yıl 2020, , 21 - 33, 29.02.2020
https://doi.org/10.13069/jacodesmath.645018

Öz

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.
Toplam 18 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Mühendislik
Bölüm Makaleler
Yazarlar

Manjit Singh 0000-0003-3351-7287

Yayımlanma Tarihi 29 Şubat 2020
Yayımlandığı Sayı Yıl 2020

Kaynak Göster

APA Singh, M. (2020). A class of constacyclic codes containing formally self-dual and isodual codes. Journal of Algebra Combinatorics Discrete Structures and Applications, 7(1), 21-33. https://doi.org/10.13069/jacodesmath.645018
AMA Singh M. A class of constacyclic codes containing formally self-dual and isodual codes. Journal of Algebra Combinatorics Discrete Structures and Applications. Şubat 2020;7(1):21-33. doi:10.13069/jacodesmath.645018
Chicago Singh, Manjit. “A Class of Constacyclic Codes Containing Formally Self-Dual and Isodual Codes”. Journal of Algebra Combinatorics Discrete Structures and Applications 7, sy. 1 (Şubat 2020): 21-33. https://doi.org/10.13069/jacodesmath.645018.
EndNote Singh M (01 Şubat 2020) A class of constacyclic codes containing formally self-dual and isodual codes. Journal of Algebra Combinatorics Discrete Structures and Applications 7 1 21–33.
IEEE M. Singh, “A class of constacyclic codes containing formally self-dual and isodual codes”, Journal of Algebra Combinatorics Discrete Structures and Applications, c. 7, sy. 1, ss. 21–33, 2020, doi: 10.13069/jacodesmath.645018.
ISNAD Singh, Manjit. “A Class of Constacyclic Codes Containing Formally Self-Dual and Isodual Codes”. Journal of Algebra Combinatorics Discrete Structures and Applications 7/1 (Şubat 2020), 21-33. https://doi.org/10.13069/jacodesmath.645018.
JAMA Singh M. A class of constacyclic codes containing formally self-dual and isodual codes. Journal of Algebra Combinatorics Discrete Structures and Applications. 2020;7:21–33.
MLA Singh, Manjit. “A Class of Constacyclic Codes Containing Formally Self-Dual and Isodual Codes”. Journal of Algebra Combinatorics Discrete Structures and Applications, c. 7, sy. 1, 2020, ss. 21-33, doi:10.13069/jacodesmath.645018.
Vancouver Singh M. A class of constacyclic codes containing formally self-dual and isodual codes. Journal of Algebra Combinatorics Discrete Structures and Applications. 2020;7(1):21-33.