Construction of Quasi-Twisted Codes and Enumeration of Defining Polynomials

  • Aaron Gulliver
  • Vadlamudi Ch. Venkaiah
Keywords: Finite fields, Twistulant matrices, Quasi-twisted codes, Optimal codes, Griesmer bound


Let $d_{q}(n,k)$ be the maximum possible minimum Hamming distance of a linear [$n,k$] code over $\F_{q}$. 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 $\FF_q$ is enumerated and linear codes over $\F_{17}$ and $\F_{19}$ are constructed for $k$ up to $5$.


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.

W. Bosma, J. Cannon, C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput., 24(3-4) (1997) 235–265.

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.

E. Z. Chen, N. Aydin, A database of linear codes over F 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.

J. A. Gallian, Contemporary Abstract Algebra, Eighth Edition, Brooks/Cole, Boston, MA 2013.

M. Grassl, Code Tables: Bounds on the parameters of various types of codes, available online at

P.P. Greenough, R. Hill, Optimal ternary quasi-cyclic codes, Des. Codes, Cryptogr. 2(1) (1992) 81–91.

T. A. Gulliver, Quasi-twisted codes over F 11 , Ars Combinatoria 99 (2011) 3–17.

T. A. Gulliver, New optimal ternary linear codes, IEEE Trans. Inform. Theory 41(4) (1995), 1182–1185.

T. A. Gulliver, V. K. Bhargava, Some 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.

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.

F. J. MacWilliams, N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, New York, NY 1977.

D. W. Newhart, On minimum weight codewords in QR codes, J. Combin. Theory Ser. A 48(1) (1988) 104–119.

V. Ch. Venkaiah, T. A. Gulliver, Quasi-cyclic codes over F 13 and enumeration of defining polynomials, J. Discrete Algorithms 16 (2012) 249–257.