Year 2020, Volume 7 , Issue 1, Pages 3 - 20 2020-02-29

Construction of quasi-twisted codes and enumeration of defining polynomials

T. Aaron GULLİVER [1] , Vadlamudi Ch. VENKAİAH [2]


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 $5$.
Finite fields, Twistulant matrices, Quasi-twisted codes, Optimal codes, Griesmer bound
  • [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.
Primary Language en
Subjects Engineering
Journal Section Articles
Authors

Author: T. Aaron GULLİVER (Primary Author)

Author: Vadlamudi Ch. VENKAİAH

Dates

Publication Date : February 29, 2020

Bibtex @research article { jacodesmath645015, journal = {Journal of Algebra Combinatorics Discrete Structures and Applications}, issn = {}, eissn = {2148-838X}, address = {}, publisher = {Yildiz Technical University}, year = {2020}, volume = {7}, pages = {3 - 20}, doi = {10.13069/jacodesmath.645015}, title = {Construction of quasi-twisted codes and enumeration of defining polynomials}, key = {cite}, author = {Gulli̇ver, T. Aaron and Venkai̇ah, Vadlamudi Ch.} }
APA Gulli̇ver, T , Venkai̇ah, V . (2020). Construction of quasi-twisted codes and enumeration of defining polynomials . Journal of Algebra Combinatorics Discrete Structures and Applications , 7 (1) , 3-20 . DOI: 10.13069/jacodesmath.645015
MLA Gulli̇ver, T , Venkai̇ah, V . "Construction of quasi-twisted codes and enumeration of defining polynomials" . Journal of Algebra Combinatorics Discrete Structures and Applications 7 (2020 ): 3-20 <https://dergipark.org.tr/en/pub/jacodesmath/issue/51990/645015>
Chicago Gulli̇ver, T , Venkai̇ah, V . "Construction of quasi-twisted codes and enumeration of defining polynomials". Journal of Algebra Combinatorics Discrete Structures and Applications 7 (2020 ): 3-20
RIS TY - JOUR T1 - Construction of quasi-twisted codes and enumeration of defining polynomials AU - T. Aaron Gulli̇ver , Vadlamudi Ch. Venkai̇ah Y1 - 2020 PY - 2020 N1 - doi: 10.13069/jacodesmath.645015 DO - 10.13069/jacodesmath.645015 T2 - Journal of Algebra Combinatorics Discrete Structures and Applications JF - Journal JO - JOR SP - 3 EP - 20 VL - 7 IS - 1 SN - -2148-838X M3 - doi: 10.13069/jacodesmath.645015 UR - https://doi.org/10.13069/jacodesmath.645015 Y2 - 2019 ER -
EndNote %0 Journal of Algebra Combinatorics Discrete Structures and Applications Construction of quasi-twisted codes and enumeration of defining polynomials %A T. Aaron Gulli̇ver , Vadlamudi Ch. Venkai̇ah %T Construction of quasi-twisted codes and enumeration of defining polynomials %D 2020 %J Journal of Algebra Combinatorics Discrete Structures and Applications %P -2148-838X %V 7 %N 1 %R doi: 10.13069/jacodesmath.645015 %U 10.13069/jacodesmath.645015
ISNAD Gulli̇ver, T. Aaron , Venkai̇ah, 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
AMA Gulli̇ver T , Venkai̇ah V . Construction of quasi-twisted codes and enumeration of defining polynomials. Journal of Algebra Combinatorics Discrete Structures and Applications. 2020; 7(1): 3-20.
Vancouver Gulli̇ver T , Venkai̇ah V . Construction of quasi-twisted codes and enumeration of defining polynomials. Journal of Algebra Combinatorics Discrete Structures and Applications. 2020; 7(1): 3-20.

Authors of the Article
T. Aaron GULLİVER [1]
Vadlamudi Ch. VENKAİAH [2]