Abstract
Let n q (k, d) denote the smallest value of n for which there exists a linear [n, k, d]-code over GF(q). An [n, k, d]-code whose length is equal to n q (k, d) is called optimal. The problem of finding n q (k, d)has received much attention for the case q = 2. We generalize several results to the case of an arbitrary prime power q as well as introducing new results and a detailed methodology to enable the problem to be tackled over any finite field.
In particular, we study the problem with q = 3 and determine n 3(k, d) for all d when k ≤ 4, and n 3(5, d) for all but 30 values of d.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Belov, B.I., Logachev, V.N., and Sandimirov, V.P. 1974. Construction of a class of linear binary codes achieving the Varshimov-Griesmer bound. Problems of Info. transmission 10 (3):211–217.
Dodunekov, S.M. 1984. Minimum block length of a linear q-ary code with specified dimension and code distance. Problems of Info. Trans. 20:239–249.
Dodunekov, S.M., Helleseth, T., Manev, N., and Ytrehus, φ. 1987. New bounds on binary linear codes of dimension eight. IEEE Trans. Info. Theory 33:917–919.
Farrell, P.G. 1970. Linear binary anticodes. Electronics Letters 6:419–421.
Games, R.A. 1983. The packing problem for projective geometries over GF(3) with dimension greater than 5 J. Comb. Theory, Series A, 35:126–144.
Greenough, P.P., and Hill, R. 1992. Optimal ternary quasi-cyclic codes. Designs, Codes and Cryptography 2:81–91.
Griesmer, J.H. 1960. A bound for error-correcting codes. I.B.M. J. Res. Develop 4:532–542.
Hamada, N., and Deza, M. 1989. A survey of recent works with respect to a characterization of an (n, k, d; q)-code meeting the Griesmer bound using a min. hyper in a finite projective geometry. Discrete Math. 77:75–87.
Helgert, H.J., and Stinaff, R.D. 1973. Minimum-distance bounds for binary linear codes. IEEE Trans. Info. Theory 19:344–356.
Helleseth T., and Ytrehus, φ. 1986. New bounds on the minimum length of binary linear block codes of dimension 8. Report on Information, no. 21, Dept. of Informatics, Univ. of Bergen, Norway.
Hill, R. 1978. Caps and codes. Discrete Math. 22:111–137.
Hill, R. 1991. Optimal linear codes. In Cryptography and Coding II, edited by C. Mitchell, Oxford University Press, 75–104.
Hill, R., and Newton, D.E. 1988. Some optimal ternary linear codes. Ars Combinatoria, 25A:61–72.
MacWilliams, F.J., and Sloane, N.J. A. 1977. The theory of error-correcting codes, Amsterdam: North-Holland.
Newton, D.E. 1990. Optimal ternary linear codes. Ph.D. Thesis, Department of Mathematics and Computer Science, University of Salford.
Pless, V. 1982. Introduction to the theory of error-correcting codes, New York: Wiley.
Solomon, G., and Stiffler, J.J. 1965. Algebraically punctured cyclic codes. Info. and Control 8:170–179.
van Tilborg, H.C.A. 1981. The smallest length of binary 7-dimensional linear codes with prescribed minimum distance. Disc. Math. 33:197–207.
Verhoeff, T.. 1987. An updated table of minimum-distance bounds for binary linear codes. IEEE Trans. Info. Theory, Vol. IT-33:665–680.
Ytrehus, φ., and Helleseth, T. 1990. There is no binary [25, 8, 10]-code. IEEE Trans. Info Theory, 36:695–696.
Author information
Authors and Affiliations
Additional information
Communicated by H. van Tilborg
Rights and permissions
About this article
Cite this article
Hill, R., Newton, D.E. Optimal ternary linear codes. Des Codes Crypt 2, 137–157 (1992). https://doi.org/10.1007/BF00124893
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00124893