Construction of quasi-twisted codes and enumeration of defining polynomials

Year 2020, Volume: 7 Issue: 1 (Special Issue in Algebraic Coding Theory: New Trends and Its Connections), 3 - 20, 29.02.2020

T. Aaron Gulliver Vadlamudi Ch. Venkaiah

https://doi.org/10.13069/jacodesmath.645015

Cited By: 1

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 $5$.

Keywords

Finite fields, Twistulant matrices, Quasi-twisted codes, Optimal codes, Griesmer bound

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.