New results and bounds on codes over $GF(17)$
DOI:
https://doi.org/10.13069/jacodesmath.v11i1.257Keywords:
Quasi-cyclic codes, Quasi-twisted codes, Search algorithms for linear codes, Distance bounds, Lower bounds, Upper bounds, Optimal codesAbstract
Determining the best possible values of the parameters of a linear code is one of the most fundamental and challenging problems in coding theory. There exist databases of best-known linear codes (BKLC) over small finite fields. In this work, we establish a database of BKLCs over the field $GF(17)$ together with upper bounds on the minimum distances for lengths up to $150$ and dimensions up to $6$. In the process, we have found many new linear codes over $GF(17)$.
Received: 10 August 2022 | Accepted: 5 December 2022Downloads
References
D. Akre, N. Aydin, M. Harrington, and S. Pandey, New binary and ternary quasi-cyclic codes with good properties, Comput. Appl. Math. 42(2) (2023) Article 102.
N. Aydin, I. Siap, and D. K. Ray-Chaudhuri, The structure of 1-generator quasi-twisted codes and new linear codes, Des. Codes Cryptogr. 24(3) (2001) 313–326.
N. Aydin and J. Murphree, New linear codes from constacyclic codes, J. Franklin Inst. 351(3) (2014) 1691–1699.
S. Ball, Table of bounds on three dimensional linear codes or $(n,r)$-arcs in $PG(2,q)$, Online Available at: https://web.mat.upc.edu/simeon.michael.ball/codebounds.html.
E. R. Berlekamp, Algebraic Coding Theory, Aegean Park Press, Walnut Creek, 1984.
K. Betsumiya, S. Georgiou, T. A. Gulliver, M. Harada, and C. Koukouvinos, On self-dual codes over some prime fields, Discrete Math. 262(1-3) (2003) 37–58.
W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24(3-4) (1997) 235–265.
E. Z. Chen, Six new binary quasi-cyclic codes, IEEE Trans. Inform. Theory 40(5) (1994) 1666–1667.
E. Z. Chen, A new iterative computer search algorithm for good quasi-twisted codes, Des. Codes Cryptogr. 76(2) (2015) 307–323.
E. Z. Chen, Online Database of Quasi-Twisted Codes, Available at: http://databases.cs.hkr.se/qtcodes/index.htm.
E. Z. Chen and N. Aydin, A database of linear codes over $\mathbb{F}_{13}$ with minimum distance bounds and new quasi-twisted codes from a heuristic search algorithm, J. Algebra Comb. Discrete Appl. 2(1) (2015) 1–16.
E. Z. Chen and N. Aydin, New quasi-twisted codes over $\mathbb{F}_{11}$ - minimum distance bounds and a new database, J. Inform. Optim. Sci. 36(1-2) (2015) 129–157.
J. Chen, K. W. Shum, Q. Yu, and C. W. Sung, Sector-disk codes and partial MDS codes with up to three global parities, in: 2015 IEEE International Symposium on Information Theory (ISIT), Hong Kong, 2015, pp. 1876–1880.
R. N. Daskalov, T. A. Gulliver, and E. Metodieva, New good quasi-cyclic ternary and quaternary linear codes, IEEE Trans. Inform. Theory 43(5) (1997) 1647–1650.
R. N. Daskalov, T. A. Gulliver, and E. Metodieva, New ternary linear codes, IEEE Trans. Inform. Theory 45(5) (1999) 1687–1688.
M. Grassl, Bounds on the minimum distance of linear codes and quantum codes, Online available at: http://www.codetables.de.
T. A. Gulliver and V. K. Bhargava, Some best rate $1/p$ and rate $(p-1)/p$ systematic quasi-cyclic codes, IEEE Trans. Inform. Theory 37(2) (1991) 552–555.
T. A. Gulliver and P. R. J. Östergård, Improved bounds for ternary linear codes of dimension 7, IEEE Trans. Inform. Theory 43(4) (1997) 1377–1388.
T. A. Gulliver and V. C. Venkaiah, Construction of quasi-twisted codes and enumeration of defining polynomials, J. Algebra Comb. Discrete Appl. 7(1) (2020) 3–20.
R. Hill, Optimal linear codes, in: Cryptography and Coding II, C. Mitchell (Ed.), Oxford University Press, 1992, pp. 75–104.
I. S. Kotsireas, C. Koukouvinos, and D. E. Simos, MDS and near-MDS self-dual codes over large prime fields, Adv. Math. Commun. 3(4) (2009) 349–361.
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland Publishing Company, Amsterdam, 1977.
A. Vardy, The intractability of computing the minimum distance of a code, IEEE Trans. Inform. Theory 43(6) (1997) 1757–1766.
