×

The nonexistence of ternary \([105,6,68]\) and \([230,6,152]\) codes. (English) Zbl 1057.94024

Summary: Let \([n,k,d]_q\)-codes be linear codes of length \(n\), dimension \(k\) and minimum Hamming distance \(d\) over GF\((q)\). The nonexistence of \([105,6,68]_3\) and \([230,6,152]_3\) codes is proved.

MSC:

94B05 Linear codes (general theory)
Full Text: DOI

References:

[1] G. Bogdanova, I. Boukliev, New linear codes of dimension 5 over \(\operatorname{GF}(3)\); G. Bogdanova, I. Boukliev, New linear codes of dimension 5 over \(\operatorname{GF}(3)\)
[2] Boukliev, I., Some new optimal ternary linear codes, Designs Codes Cryptogr., 12, 5-11 (1997) · Zbl 0903.94046
[3] Bouyukliev, I.; Simonis, J., Some new results for optimal ternary linear codes, IEEE Trans. Inform. Theory, 48, 4, 981-985 (2002) · Zbl 1061.94061
[4] A.E. Brouwer, Minimum distance bounds for linear codes over \(\operatorname{GF}(q)(q = 2, 3, 4, 5, 7, 8, 9)\); A.E. Brouwer, Minimum distance bounds for linear codes over \(\operatorname{GF}(q)(q = 2, 3, 4, 5, 7, 8, 9)\)
[5] Daskalov, R. N., (Bounds on the minimum length for ternary linear codes of dimension six (1993), Mathematics and Education in Mathematics: Mathematics and Education in Mathematics Sofia), 15-22
[6] Dodunekov, S. M., Minimum block length of a linear \(q\)-ary code with specified dimension and code distance, Probl. Inform. Transm., 20, 239-249 (1984) · Zbl 0568.94024
[7] van Eupen, M., Five new optimal ternary linear codes, IEEE Trans. Inform. Theory, 40, 193 (1994) · Zbl 0802.94016
[8] van Eupen, M., Some new results for ternary linear codes of dimension 5 and 6, IEEE Trans. Inform. Theory, 41, 2048-2051 (1995) · Zbl 0847.94008
[9] van Eupen, M.; Hamada, N.; Watamori, Y., The nonexistence of ternary \([50, 5, 32]\) codes, Designs Codes Cryptogr., 7, 235-237 (1996) · Zbl 0843.94015
[10] van Eupen, M.; Hill, R., An optimal ternary \([69, 5, 45]\) code and related codes, Designs Codes Cryptogr., 4, 271-282 (1994) · Zbl 0817.94020
[11] van Eupen, M.; Lisonek, P., Classification of some optimal ternary linear codes of small length, Designs Codes Cryptogr., 10, 63-84 (1997) · Zbl 0869.94031
[12] Griesmer, J. H., A bound for error-correcting codes, IBM J. Res. Develop., 4, 532-542 (1960) · Zbl 0234.94009
[13] Hamada, N., A survey of recent work on characterization of minihypers in \(PG(t, q)\) and nonbinary linear codes meeting the Griesmer bound, J. Combin. Inform. System Sci., 18, 229-268 (1993)
[14] Hamada, N., The nonexistence of ternary [29,6,17] codes, Math. Japon., 46, 253-264 (1997) · Zbl 0956.94015
[15] Hamada, N.; van Eupen, M., The nonexistence of ternary [38,6,23] codes, Designs Codes Cryptogr., 13, 165-172 (1998) · Zbl 0895.94012
[16] Hamada, N.; Helleseth, T., The nonexistence of ternary [97,6,63] codes, J. Statist. Plann. Inference, 106, 485-507 (2002) · Zbl 0995.05026
[17] Hamada, N.; Helleseth, T.; Martinsen, H. M.; Ytrehus, O., There is no ternary [28,6,16] codes, IEEE Trans. Inform. Theory, 46, 4, 1550-1554 (2000) · Zbl 0998.94554
[18] Hamada, N.; Watamori, Y., The nonexistence of some ternary linear codes of dimension 6 and the bounds for \(n_3(6, d), 1 \leqslant d \leqslant 243\), Math. Japon., 43, 3, 577-593 (1996) · Zbl 0856.94032
[19] Hamada, N.; Watamori, Y., The nonexistence of \([71, 5, 46]_3\) codes, J. Statist. Plann. Inference, 52, 379-394 (1996) · Zbl 0854.94020
[20] Hamada, N.; Watamori, Y., The nonexistence of ternary \([79, 6, 51]\) codes, J. Statist. Plann. Inference, 72, 323-332 (1998) · Zbl 0929.94030
[21] Hamada, N.; Helleseth, T., The nonexistence of some ternary linear codes and update of the bounds for \(n_3(6, d), 1 \leqslant d \leqslant 243\), Math. Japon., 52, 1, 31-43 (2000) · Zbl 0971.94016
[22] Hill, R.; Jones, C., The nonexistence of ternary [47,6,29] codes, (Proceedings of the II International Workshop OC’98 (1998), Sozopol: Sozopol Bulgaria), 90-97
[23] Hill, R.; Newton, D. E., Optimal ternary linear codes, Designs Codes Cryptogr., 2, 137-157 (1992) · Zbl 0756.94008
[24] C. Jones, Optimal ternary linear codes, Ph.D. Thesis, University of Salford, 2000.; C. Jones, Optimal ternary linear codes, Ph.D. Thesis, University of Salford, 2000.
[25] Landgev, I., The nonexistence of some optimal ternary codes of dimension five, Designs Codes Cryptogr., 15, 245-258 (1998) · Zbl 0968.94010
[26] Lidl, R.; Niederreiter, H., (Finite Fields, Encyclopedia of Mathematics and its Applications, Vol. 20 (1983), Addison-Wesley Publishing Company: Addison-Wesley Publishing Company MA) · Zbl 0554.12010
[27] MacWilliams, F. J.; Sloane, N. J.A., The Theory of Error-Correcting Codes (1977), North-Holland: North-Holland Amsterdam · Zbl 0369.94008
[28] T. Maruta, Personal communication, 2002; T. Maruta, Personal communication, 2002
[29] Solomon, G.; Stiffler, J. J., Algebraically punctured cyclic codes, Inform. and Control, 8, 170-179 (1965) · Zbl 0149.15903
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.