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Construction of quasi-twisted codes and enumeration of defining polynomials

Year 2020, , 3 - 20, 29.02.2020
https://doi.org/10.13069/jacodesmath.645015

Abstract

Let $d_{q}(n,k)$ be the maximum possible minimum Hamming distance of a linear [$n,k$] code over $\mathbb{F}_{q}$.
Tables of best known linear codes exist for small fields and some results are known for larger fields.
Quasi-twisted codes are constructed using $m \times m$ twistulant matrices and many of these are the best known codes.
In this paper, the number of $m \times m$ twistulant matrices over $\mathbb{F}_q$ is enumerated
and linear codes over $\mathbb{F}_{17}$ and $\mathbb{F}_{19}$ are constructed for $k$ up to $5$.

References

  • [1] K. Betsumiya, S. Georgiou, T. A. Gulliver, M. Harada, C. Koukouvinos, On self-dual codes over some prime fields, Disc. Math. 262(1–3) (2003) 37–58.
  • [2] W. Bosma, J. Cannon, C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput., 24(3-4) (1997) 235–265.
  • [3] E. Z. Chen, N. Aydin, New quasi-twisted codes over $F_{11}$–minimum distance bounds and a new database, J. Inform. Optimization Sci., 36(1-2) (2015) 129–157.
  • [4] E. Z. Chen, N. Aydin, A database of linear codes over $\FF_{13}$ with minimum distance bounds and new quasi-twisted codes from a heuristic search algorithm, J. Algebra Comb. Discrete Appl., 2(1) (2015) 1–16.
  • [5] J. A. Gallian, Contemporary Abstract Algebra, Eighth Edition, Brooks/Cole, Boston, MA 2013.
  • [6] M. Grassl, Code Tables: Bounds on the parameters of various types of codes, available online at http://www.codetables.de.
  • [7] P.P. Greenough, R. Hill, Optimal ternary quasi-cyclic codes, Des. Codes, Cryptogr. 2(1) (1992) 81–91.
  • [8] T. A. Gulliver, Quasi-twisted codes over $F_{11}$, Ars Combinatoria 99 (2011) 3–17.
  • [9] T. A. Gulliver, New optimal ternary linear codes, IEEE Trans. Inform. Theory 41(4) (1995), 1182–1185.
  • [10] T. A. Gulliver, V. K. Bhargava, SSome best rate $1/p$ and rate $(p-1)/p$ systematic quasi-cyclic codes over $GF(3)$ and $GF(4)$, IEEE Trans. Inform. Theory 38(4) (1992) 1369–1374.
  • [11] T. A. Gulliver, V. K. Bhargava, New good rate $(m-1)/pm$ ternary and quaternary quasi-cyclic codes, Des. Codes, Cryptogr. 7(3) (1996) 223–233.
  • [12] F. J. MacWilliams, N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, New York, NY 1977.
  • [13] D. W. Newhart, On minimum weight codewords in QR codes, J. Combin. Theory Ser. A 48(1) (1988) 104–119.
  • [14] V. Ch. Venkaiah, T. A. Gulliver, Quasi-cyclic codes over $\FF_{13}$ and enumeration of defining polynomials, J. Discrete Algorithms 16 (2012) 249–257.
Year 2020, , 3 - 20, 29.02.2020
https://doi.org/10.13069/jacodesmath.645015

Abstract

References

  • [1] K. Betsumiya, S. Georgiou, T. A. Gulliver, M. Harada, C. Koukouvinos, On self-dual codes over some prime fields, Disc. Math. 262(1–3) (2003) 37–58.
  • [2] W. Bosma, J. Cannon, C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput., 24(3-4) (1997) 235–265.
  • [3] E. Z. Chen, N. Aydin, New quasi-twisted codes over $F_{11}$–minimum distance bounds and a new database, J. Inform. Optimization Sci., 36(1-2) (2015) 129–157.
  • [4] E. Z. Chen, N. Aydin, A database of linear codes over $\FF_{13}$ with minimum distance bounds and new quasi-twisted codes from a heuristic search algorithm, J. Algebra Comb. Discrete Appl., 2(1) (2015) 1–16.
  • [5] J. A. Gallian, Contemporary Abstract Algebra, Eighth Edition, Brooks/Cole, Boston, MA 2013.
  • [6] M. Grassl, Code Tables: Bounds on the parameters of various types of codes, available online at http://www.codetables.de.
  • [7] P.P. Greenough, R. Hill, Optimal ternary quasi-cyclic codes, Des. Codes, Cryptogr. 2(1) (1992) 81–91.
  • [8] T. A. Gulliver, Quasi-twisted codes over $F_{11}$, Ars Combinatoria 99 (2011) 3–17.
  • [9] T. A. Gulliver, New optimal ternary linear codes, IEEE Trans. Inform. Theory 41(4) (1995), 1182–1185.
  • [10] T. A. Gulliver, V. K. Bhargava, SSome best rate $1/p$ and rate $(p-1)/p$ systematic quasi-cyclic codes over $GF(3)$ and $GF(4)$, IEEE Trans. Inform. Theory 38(4) (1992) 1369–1374.
  • [11] T. A. Gulliver, V. K. Bhargava, New good rate $(m-1)/pm$ ternary and quaternary quasi-cyclic codes, Des. Codes, Cryptogr. 7(3) (1996) 223–233.
  • [12] F. J. MacWilliams, N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, New York, NY 1977.
  • [13] D. W. Newhart, On minimum weight codewords in QR codes, J. Combin. Theory Ser. A 48(1) (1988) 104–119.
  • [14] V. Ch. Venkaiah, T. A. Gulliver, Quasi-cyclic codes over $\FF_{13}$ and enumeration of defining polynomials, J. Discrete Algorithms 16 (2012) 249–257.
There are 14 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

T. Aaron Gulliver This is me

Vadlamudi Ch. Venkaiah This is me

Publication Date February 29, 2020
Published in Issue Year 2020

Cite

APA Gulliver, T. A., & Venkaiah, V. C. (2020). Construction of quasi-twisted codes and enumeration of defining polynomials. Journal of Algebra Combinatorics Discrete Structures and Applications, 7(1), 3-20. https://doi.org/10.13069/jacodesmath.645015
AMA Gulliver TA, Venkaiah VC. Construction of quasi-twisted codes and enumeration of defining polynomials. Journal of Algebra Combinatorics Discrete Structures and Applications. February 2020;7(1):3-20. doi:10.13069/jacodesmath.645015
Chicago Gulliver, T. Aaron, and Vadlamudi Ch. Venkaiah. “Construction of Quasi-Twisted Codes and Enumeration of Defining Polynomials”. Journal of Algebra Combinatorics Discrete Structures and Applications 7, no. 1 (February 2020): 3-20. https://doi.org/10.13069/jacodesmath.645015.
EndNote Gulliver TA, Venkaiah VC (February 1, 2020) Construction of quasi-twisted codes and enumeration of defining polynomials. Journal of Algebra Combinatorics Discrete Structures and Applications 7 1 3–20.
IEEE T. A. Gulliver and V. C. Venkaiah, “Construction of quasi-twisted codes and enumeration of defining polynomials”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 7, no. 1, pp. 3–20, 2020, doi: 10.13069/jacodesmath.645015.
ISNAD Gulliver, T. Aaron - Venkaiah, Vadlamudi Ch. “Construction of Quasi-Twisted Codes and Enumeration of Defining Polynomials”. Journal of Algebra Combinatorics Discrete Structures and Applications 7/1 (February 2020), 3-20. https://doi.org/10.13069/jacodesmath.645015.
JAMA Gulliver TA, Venkaiah VC. Construction of quasi-twisted codes and enumeration of defining polynomials. Journal of Algebra Combinatorics Discrete Structures and Applications. 2020;7:3–20.
MLA Gulliver, T. Aaron and Vadlamudi Ch. Venkaiah. “Construction of Quasi-Twisted Codes and Enumeration of Defining Polynomials”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 7, no. 1, 2020, pp. 3-20, doi:10.13069/jacodesmath.645015.
Vancouver Gulliver TA, Venkaiah VC. Construction of quasi-twisted codes and enumeration of defining polynomials. Journal of Algebra Combinatorics Discrete Structures and Applications. 2020;7(1):3-20.