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On optimal ternary linear codes of dimension 6

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  • We prove that $[g_3(6,d),6,d]_3$ codes for $d=253$-$267$ and $[g_3(6,d)+1,6,d]_3$ codes for $d=302, 303, 307$-$312$ exist and that $[g_3(6,d),6,d]_3$ codes for $d=175, 200, 302, 303, 308, 309$ and a $[g_3(6,133)+1,6,133]_3$ code do not exist, where $g_3(k,d)=\sum_{i=0}^{k-1} \lceil d / 3^i \rceil$. These determine $n_3(6,d)$ for $d=133, 175, 200, 253$-$267, 302, 303, 308$-$312$, where $n_q(k,d)$ is the minimum length $n$ for which an $[n,k,d]_q$ code exists. The updated $n_3(6,d)$ table is also given.
    Mathematics Subject Classification: Primary: 94B27, 94B05; Secondary: 51E20, 05B25.


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  • [1]

    A. Beutelspacher, Blocking sets and partial spreads in finite projective spaces, Geom. Dedicata, 9 (1980), 425-449.doi: 10.1007/BF00181559.


    R. C. Bose and R. C. Burton, A characterization of flat spaces in a finite projective geometry and the uniqueness of the Hamming and the MacDonald codes, J. Combin. Theory, 1 (1966), 96-104.doi: 10.1016/S0021-9800(66)80007-8.


    A. A. Bruen, Polynomial multiplicities over finite fields and intersection sets, J. Combin. Theory Ser. A, 60 (1992), 19-33.doi: 10.1016/0097-3165(92)90035-S.


    M. van Eupen and R. Hill, An optimal ternary $[69,5,45]$ code and related codes, Des. Codes Cryptogr., 4 (1994), 271-282.doi: 10.1007/BF01388456.


    M. van Eupen and P. Lisonêk, Classification of some optimal ternary linear codes of small length, Des. Codes Cryptogr., 10 (1997), 63-84.doi: 10.1023/A:1008292320488.


    N. Hamada, A characterization of some $[n,k,d;q]$-codes meeting the Griesmer bound using a minihyper in a finite projective geometry, Discrete Math., 116 (1993), 229-268.doi: 10.1016/0012-365X(93)90404-H.


    N. Hamada, The nonexistence of $[303,6,201;3]$-codes meeting the Griesmer bound, Technical Report OWUAM-009, Osaka Women's University, 1995.


    N. Hamada and T. Helleseth, The uniqueness of $[87,5,57;3]$ codes and the nonexistence of $[258,6,171;3]$ codes, J. Statist. Plann. Inference, 56 (1996), 105-127.doi: 10.1016/S0378-3758(96)00013-4.


    N. Hamada and T. Helleseth, The nonexistence of some ternary linear codes and update of the bounds for $n_3(6,d)$, $1\leq d\leq 243$, Math. Japon., 52 (2000), 31-43.


    N. Hamada and T. Maruta, A survey of recent results on optimal linear codes and minihypers, unpublished manuscript, 2003.


    U. Heim, On $t$-blocking sets in projective spaces, unpublished manuscript, 1994.


    R. Hill, Caps and codes, Discrete Math., 22 (1978), 111-137.doi: 10.1016/0012-365X(78)90120-6.


    R. Hill, Optimal linear codes, in "Cryptography and Coding II'' (ed. C. Mitchell), Oxford Univ. Press, Oxford, (1992), 75-104.


    R. Hill, An extension theorem for linear codes, Des. Codes Cryptogr., 17 (1999), 151-157.doi: 10.1023/A:1008319024396.


    R. Hill and D. E. Newton, Optimal ternary linear codes, Des. Codes Cryptogr., 2 (1992), 137-157.doi: 10.1007/BF00124893.


    W. C. Huffman and V. Pless, "Fundamentals of Error-Correcting Codes," Cambridge University Press, Cambridge, 2003.


    C. M. Jones, "Optimal Ternary Linear Codes," Ph.D thesis, University of Salford, 2000.


    I. N. Landjev, The nonexistence of some optimal ternary linear codes of dimension five, Des. Codes Cryptogr., 15 (1998), 245-258.doi: 10.1023/A:1008317124941.


    I. Landgev, T. Maruta and R. Hill, On the nonexistence of quaternary $[51,4,37]$ codes, Finite Fields Appl., 2 (1996), 96-110.


    T. Maruta, On the nonexistence of $q$-ary linear codes ofdimension five, Des. Codes Cryptogr., 22 (2001), 165-177.


    T. Maruta, Extendability of ternary linear codes, Des. Codes Cryptogr., 35 (2005), 175-190.doi: 10.1007/s10623-005-6400-7.


    T. Maruta, "Griesmer Bound for Linear Codes over Finite Fields," available online at http://www.geocities.jp/mars39geo/griesmer.htm


    Y. OyaThe nonexistence of $[132,6,86]_3$ codes and $[135,6,88]_3$ codes, Serdica J. Comput., to appear.


    G. Pellegrino, Sul massimo ordine delle calotte in S4,3, Matematiche (Catania), 25 (1970), 1-9.


    M. Takenaka, K. Okamoto and T. Maruta, On optimal non-projective ternary linear codes, Discrete Math., 308 (2008), 842-854.doi: 10.1016/j.disc.2007.07.044.


    H. N. Ward, Divisibility of codes meeting the Griesmer bound, J. Combin. Theory Ser. A, 83 (1998), 79-93.doi: 10.1006/jcta.1997.2864.


    Y. Yoshida and T. Maruta, Ternary linear codes and quadrics, Electronic J. Combin., 16 (2009), 21 pp.

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