August  2011, 5(3): 505-520. doi: 10.3934/amc.2011.5.505

On optimal ternary linear codes of dimension 6

1. 

Department of Mathematics and Information Sciences, Osaka Prefecture University, Sakai, Osaka 599-8531, Japan

Received  November 2010 Revised  April 2011 Published  August 2011

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.
Citation: Tatsuya Maruta, Yusuke Oya. On optimal ternary linear codes of dimension 6. Advances in Mathematics of Communications, 2011, 5 (3) : 505-520. doi: 10.3934/amc.2011.5.505
References:
[1]

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

[2]

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.  doi: 10.1016/S0021-9800(66)80007-8.  Google Scholar

[3]

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

[4]

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

[5]

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

[6]

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.  doi: 10.1016/0012-365X(93)90404-H.  Google Scholar

[7]

N. Hamada, The nonexistence of $[303,6,201;3]$-codes meeting the Griesmer bound,, Technical Report OWUAM-009, (1995).   Google Scholar

[8]

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.  doi: 10.1016/S0378-3758(96)00013-4.  Google Scholar

[9]

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.   Google Scholar

[10]

N. Hamada and T. Maruta, A survey of recent results on optimal linear codes and minihypers,, unpublished manuscript, (2003).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

[11]

U. Heim, On $t$-blocking sets in projective spaces,, unpublished manuscript, (1994).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

[12]

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

[13]

R. Hill, Optimal linear codes,, in, (1992), 75.   Google Scholar

[14]

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

[15]

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

[16]

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

[17]

C. M. Jones, "Optimal Ternary Linear Codes,", Ph.D thesis, (2000).   Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

T. Maruta, "Griesmer Bound for Linear Codes over Finite Fields,", available online at \url{http://www.geocities.jp/mars39geo/griesmer.htm}, ().   Google Scholar

[23]

Y. Oya, The nonexistence of $[132,6,86]_3$ codes and $[135,6,88]_3$ codes,, Serdica J. Comput., ().   Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

show all references

References:
[1]

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

[2]

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.  doi: 10.1016/S0021-9800(66)80007-8.  Google Scholar

[3]

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

[4]

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

[5]

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

[6]

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.  doi: 10.1016/0012-365X(93)90404-H.  Google Scholar

[7]

N. Hamada, The nonexistence of $[303,6,201;3]$-codes meeting the Griesmer bound,, Technical Report OWUAM-009, (1995).   Google Scholar

[8]

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.  doi: 10.1016/S0378-3758(96)00013-4.  Google Scholar

[9]

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.   Google Scholar

[10]

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

[11]

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

[12]

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

[13]

R. Hill, Optimal linear codes,, in, (1992), 75.   Google Scholar

[14]

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

[15]

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

[16]

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

[17]

C. M. Jones, "Optimal Ternary Linear Codes,", Ph.D thesis, (2000).   Google Scholar

[18]

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

[19]

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

[20]

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

[21]

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

[22]

T. Maruta, "Griesmer Bound for Linear Codes over Finite Fields,", available online at \url{http://www.geocities.jp/mars39geo/griesmer.htm}, ().   Google Scholar

[23]

Y. Oya, The nonexistence of $[132,6,86]_3$ codes and $[135,6,88]_3$ codes,, Serdica J. Comput., ().   Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

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