doi: 10.3934/amc.2021052
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Nonexistence of some ternary linear codes with minimum weight -2 modulo 9

Department of Mathematical Sciences, Osaka Prefecture University, Sakai, Osaka 599-8531, Japan

* Corresponding author: Tatsuya Maruta

Received  May 2021 Revised  September 2021 Early access November 2021

Fund Project: The second author is partially supported by JSPS KAKENHI Grant Number 20K03722

One of the fundamental problems in coding theory is to find $ n_q(k,d) $, the minimum length $ n $ for which a linear code of length $ n $, dimension $ k $, and the minimum weight $ d $ over the field of order $ q $ exists. The problem of determining the values of $ n_q(k,d) $ is known as the optimal linear codes problem. Using the geometric methods through projective geometry and a new extension theorem given by Kanda (2020), we determine $ n_3(6,d) $ for some values of $ d $ by proving the nonexistence of linear codes with certain parameters.

Citation: Toshiharu Sawashima, Tatsuya Maruta. Nonexistence of some ternary linear codes with minimum weight -2 modulo 9. Advances in Mathematics of Communications, doi: 10.3934/amc.2021052
References:
[1]

I. G. Bouyukliev, What is Q-Extension?, Serdica J. Computing, 1 (2007), 115-130.   Google Scholar

[2]

I. Bouyukliev and J. Simonis, Some new results for optimal ternary linear codes, IEEE Trans. Inform. Theory, 48 (2002), 981-985.  doi: 10.1109/18.992814.  Google Scholar

[3]

R. Daskalov and E. Metodieva, The nonexistence of ternary $[284, 6,188]$ codes, Probl. Inform. Trans., 40 (2004), 135-146.  doi: 10.1023/B:PRIT.0000043927.19508.8b.  Google Scholar

[4]

R. Daskalov and E. Metodieva, The nonexistence of ternary [105, 6, 68] and [230, 6,152] codes, Discrete Math., 286 (2004), 225-232.  doi: 10.1016/j.disc.2004.06.002.  Google Scholar

[5]

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.  Google Scholar

[6]

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.  Google Scholar

[7]

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

[8]

R. Hill, Optimal linear codes, Cryptography and Coding, 33 (1992), 75-104.   Google Scholar

[9]

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

[10]

R. Hill and P. Lizak, Extensions of linear codes, Proc. IEEE Int. Symposium on Inform. Theory, (1995), 345. doi: 10.1109/ISIT.1995.550332.  Google Scholar

[11]

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

[12] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.  Google Scholar
[13]

C. M. Jones, Optimal Ternary Linear Codes, PhD thesis, University of Salford, 2000. Google Scholar

[14]

H. Kanda, A new extension theorem for ternary linear codes and its application, Finite Fields Appl., 67 (2020), 101711.  doi: 10.1016/j.ffa.2020.101711.  Google Scholar

[15]

K. Kumegawa, T. Okazaki and T. Maruta, On the minimum length of linear codes over the field of 9 elements, Electron. J. Combin., 24 (2017), #P1.50. doi: 10.37236/6394.  Google Scholar

[16]

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.  Google Scholar

[17]

I. N. Landjev and T. Maruta, On the minimum length of quaternary linear codes of dimension five, Discrete Math., 202 (1999), 145-161.  doi: 10.1016/S0012-365X(98)00354-9.  Google Scholar

[18]

I. LandgevT. Maruta and R. Hill, On the nonexistence of quaternary $[51, 4, 37]$ codes, Finite Fields Appl., 2 (1996), 96-110.  doi: 10.1006/ffta.1996.0007.  Google Scholar

[19]

I. Landjev and P. Vandendriessche, A study of (xvt; xvt−1)-minihypers in PG(t, q), J. Combin. Theory Ser. A, 119 (2012), 1123-1131.  doi: 10.1016/j.jcta.2012.02.009.  Google Scholar

[20]

T. Maruta, On the achievement of the Griesmer bound, Des. Codes Cryptogr., 12 (1997), 83-87.  doi: 10.1023/A:1008250010928.  Google Scholar

[21]

T. Maruta, On the nonexistence of $q$-ary linear codes of dimension five, Des. Codes Cryptogr., 22 (2001), 165-177.  doi: 10.1023/A:1008317022638.  Google Scholar

[22]

T. Maruta, The nonexistence of some ternary linear codes of dimension 6, Discrete Math., 288 (2004), 125-133.  doi: 10.1016/j.disc.2004.07.003.  Google Scholar

[23]

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

[24]

T. Maruta, Griesmer bound for linear codes over finite fields, Available from: http://mars39.lomo.jp/opu/griesmer.htm. Google Scholar

[25]

T. Maruta and K. Okamoto, Some improvements to the extendability of ternary linear codes, Finite Fields Appl., 13 (2007), 259-280.  doi: 10.1016/j.ffa.2005.09.005.  Google Scholar

[26]

T. Maruta and Y. Oya, On optimal ternary linear codes of dimension 6, Adv. Math. Commun., 5 (2011), 505-520.  doi: 10.3934/amc.2011.5.505.  Google Scholar

[27]

T. Maruta and Y. Oya, On the minimum length of ternary linear codes, Des. Codes Cryptogr., 68 (2013), 407-425.  doi: 10.1007/s10623-011-9593-y.  Google Scholar

[28]

T. Sawashima and T. Maruta, Nonexistence of some ternary linear codes, Discrete Math., 344 (2021), 112572.  doi: 10.1016/j.disc.2021.112572.  Google Scholar

[29]

M. TakenakaK. 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.  Google Scholar

[30]

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.  Google Scholar

[31]

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.  Google Scholar

[32]

Y. Yoshida and T. Maruta, Ternary linear codes and quadrics, Electronic J. Combin., 16 (2009), #R9, 21pp. doi: 10.37236/98.  Google Scholar

show all references

References:
[1]

I. G. Bouyukliev, What is Q-Extension?, Serdica J. Computing, 1 (2007), 115-130.   Google Scholar

[2]

I. Bouyukliev and J. Simonis, Some new results for optimal ternary linear codes, IEEE Trans. Inform. Theory, 48 (2002), 981-985.  doi: 10.1109/18.992814.  Google Scholar

[3]

R. Daskalov and E. Metodieva, The nonexistence of ternary $[284, 6,188]$ codes, Probl. Inform. Trans., 40 (2004), 135-146.  doi: 10.1023/B:PRIT.0000043927.19508.8b.  Google Scholar

[4]

R. Daskalov and E. Metodieva, The nonexistence of ternary [105, 6, 68] and [230, 6,152] codes, Discrete Math., 286 (2004), 225-232.  doi: 10.1016/j.disc.2004.06.002.  Google Scholar

[5]

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.  Google Scholar

[6]

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.  Google Scholar

[7]

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

[8]

R. Hill, Optimal linear codes, Cryptography and Coding, 33 (1992), 75-104.   Google Scholar

[9]

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

[10]

R. Hill and P. Lizak, Extensions of linear codes, Proc. IEEE Int. Symposium on Inform. Theory, (1995), 345. doi: 10.1109/ISIT.1995.550332.  Google Scholar

[11]

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

[12] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.  Google Scholar
[13]

C. M. Jones, Optimal Ternary Linear Codes, PhD thesis, University of Salford, 2000. Google Scholar

[14]

H. Kanda, A new extension theorem for ternary linear codes and its application, Finite Fields Appl., 67 (2020), 101711.  doi: 10.1016/j.ffa.2020.101711.  Google Scholar

[15]

K. Kumegawa, T. Okazaki and T. Maruta, On the minimum length of linear codes over the field of 9 elements, Electron. J. Combin., 24 (2017), #P1.50. doi: 10.37236/6394.  Google Scholar

[16]

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.  Google Scholar

[17]

I. N. Landjev and T. Maruta, On the minimum length of quaternary linear codes of dimension five, Discrete Math., 202 (1999), 145-161.  doi: 10.1016/S0012-365X(98)00354-9.  Google Scholar

[18]

I. LandgevT. Maruta and R. Hill, On the nonexistence of quaternary $[51, 4, 37]$ codes, Finite Fields Appl., 2 (1996), 96-110.  doi: 10.1006/ffta.1996.0007.  Google Scholar

[19]

I. Landjev and P. Vandendriessche, A study of (xvt; xvt−1)-minihypers in PG(t, q), J. Combin. Theory Ser. A, 119 (2012), 1123-1131.  doi: 10.1016/j.jcta.2012.02.009.  Google Scholar

[20]

T. Maruta, On the achievement of the Griesmer bound, Des. Codes Cryptogr., 12 (1997), 83-87.  doi: 10.1023/A:1008250010928.  Google Scholar

[21]

T. Maruta, On the nonexistence of $q$-ary linear codes of dimension five, Des. Codes Cryptogr., 22 (2001), 165-177.  doi: 10.1023/A:1008317022638.  Google Scholar

[22]

T. Maruta, The nonexistence of some ternary linear codes of dimension 6, Discrete Math., 288 (2004), 125-133.  doi: 10.1016/j.disc.2004.07.003.  Google Scholar

[23]

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

[24]

T. Maruta, Griesmer bound for linear codes over finite fields, Available from: http://mars39.lomo.jp/opu/griesmer.htm. Google Scholar

[25]

T. Maruta and K. Okamoto, Some improvements to the extendability of ternary linear codes, Finite Fields Appl., 13 (2007), 259-280.  doi: 10.1016/j.ffa.2005.09.005.  Google Scholar

[26]

T. Maruta and Y. Oya, On optimal ternary linear codes of dimension 6, Adv. Math. Commun., 5 (2011), 505-520.  doi: 10.3934/amc.2011.5.505.  Google Scholar

[27]

T. Maruta and Y. Oya, On the minimum length of ternary linear codes, Des. Codes Cryptogr., 68 (2013), 407-425.  doi: 10.1007/s10623-011-9593-y.  Google Scholar

[28]

T. Sawashima and T. Maruta, Nonexistence of some ternary linear codes, Discrete Math., 344 (2021), 112572.  doi: 10.1016/j.disc.2021.112572.  Google Scholar

[29]

M. TakenakaK. 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.  Google Scholar

[30]

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.  Google Scholar

[31]

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.  Google Scholar

[32]

Y. Yoshida and T. Maruta, Ternary linear codes and quadrics, Electronic J. Combin., 16 (2009), #R9, 21pp. doi: 10.37236/98.  Google Scholar

Table 1.  $ p_{i,j} $ for $ (\varphi_0,\varphi_1) \in \mathcal{D}_5^+ $
$ \varphi_0 $ $ \varphi_1 $ $ p_{4,0} $ $ p_{1,3} $ $ p_{1,0} $ $ p_{2,1} $
$ 40 $ $ 27 $ $ 13 $ $ 9 $ $ 18 $ $ 0 $
$ 4 $ $ 0 $ $ 9 $ $ 27 $
$ 31 $ $ 45 $ $ 10 $ $ 15 $ $ 15 $ $ 0 $
$ 1 $ $ 6 $ $ 6 $ $ 27 $
$ 40 $ $ 36 $ $ 4 $ $ 3 $ $ 6 $ $ 27 $
$ 40 $ $ 45 $ $ 4 $ $ 6 $ $ 3 $ $ 27 $
$ 49 $ $ 36 $ $ 16 $ $ 12 $ $ 12 $ $ 0 $
$ 7 $ $ 3 $ $ 3 $ $ 27 $
$ \varphi_0 $ $ \varphi_1 $ $ p_{4,0} $ $ p_{1,3} $ $ p_{1,0} $ $ p_{2,1} $
$ 40 $ $ 27 $ $ 13 $ $ 9 $ $ 18 $ $ 0 $
$ 4 $ $ 0 $ $ 9 $ $ 27 $
$ 31 $ $ 45 $ $ 10 $ $ 15 $ $ 15 $ $ 0 $
$ 1 $ $ 6 $ $ 6 $ $ 27 $
$ 40 $ $ 36 $ $ 4 $ $ 3 $ $ 6 $ $ 27 $
$ 40 $ $ 45 $ $ 4 $ $ 6 $ $ 3 $ $ 27 $
$ 49 $ $ 36 $ $ 16 $ $ 12 $ $ 12 $ $ 0 $
$ 7 $ $ 3 $ $ 3 $ $ 27 $
Table 2.  $ q_{i,j} $ for $ (\varphi_0,\varphi_1) \in \mathcal{D}_5^+ $
$ \varphi_0 $ $ \varphi_1 $ $ q_{1,3} $ $ q_{0,2} $ $ q_{2,1} $
$ 40 $ $ 27 $ $ 4 $ $ 18 $ $ 18 $
$ 31 $ $ 45 $ $ 13 $ $ 18 $ $ 9 $
$ 40 $ $ 36 $ $ 10 $ $ 15 $ $ 15 $
$ 40 $ $ 45 $ $ 16 $ $ 12 $ $ 12 $
$ 49 $ $ 36 $ $ 13 $ $ 9 $ $ 18 $
$ \varphi_0 $ $ \varphi_1 $ $ q_{1,3} $ $ q_{0,2} $ $ q_{2,1} $
$ 40 $ $ 27 $ $ 4 $ $ 18 $ $ 18 $
$ 31 $ $ 45 $ $ 13 $ $ 18 $ $ 9 $
$ 40 $ $ 36 $ $ 10 $ $ 15 $ $ 15 $
$ 40 $ $ 45 $ $ 16 $ $ 12 $ $ 12 $
$ 49 $ $ 36 $ $ 13 $ $ 9 $ $ 18 $
Table 3.  $r_{i,j}$ for $(\varphi_0,\varphi_1) \in \mathcal{D}_5^+$
$\varphi_0$ $\varphi_1$ $r_{1,0}$ $r_{0,2}$ $r_{2,1}$
40 27 22 9 9
31 45 13 18 9
40 36 16 12 12
40 45 10 15 15
49 36 13 9 18
$\varphi_0$ $\varphi_1$ $r_{1,0}$ $r_{0,2}$ $r_{2,1}$
40 27 22 9 9
31 45 13 18 9
40 36 16 12 12
40 45 10 15 15
49 36 13 9 18
Table 4.  $ p_{i,j} $ for $ (\varphi_0,\varphi_1) \in \mathcal{D}_6^+ $
$ \varphi_0 $ $ \varphi_1 $ $ p_{4,0} $ $ p_{1,3} $ $ p_{1,0} $ $ p_{2,1} $
$ 121 $ $ 81 $ $ 40 $ $ 27 $ $ 54 $ $ 0 $
$ 13 $ $ 0 $ $ 27 $ $ 81 $
$ 94 $ $ 135 $ $ 31 $ $ 45 $ $ 45 $ $ 0 $
$ 4 $ $ 18 $ $ 18 $ $ 81 $
$ 121 $ $ 108 $ $ 40 $ $ 36 $ $ 45 $ $ 0 $
$ 13 $ $ 9 $ $ 18 $ $ 81 $
$ 112 $ $ 126 $ $ 10 $ $ 15 $ $ 15 $ $ 81 $
$ 130 $ $ 117 $ $ 16 $ $ 12 $ $ 12 $ $ 81 $
$ 121 $ $ 135 $ $ 40 $ $ 45 $ $ 36 $ $ 0 $
$ 13 $ $ 18 $ $ 9 $ $ 81 $
$ 148 $ $ 108 $ $ 49 $ $ 36 $ $ 36 $ $ 0 $
$ 22 $ $ 9 $ $ 9 $ $ 81 $
$ \varphi_0 $ $ \varphi_1 $ $ p_{4,0} $ $ p_{1,3} $ $ p_{1,0} $ $ p_{2,1} $
$ 121 $ $ 81 $ $ 40 $ $ 27 $ $ 54 $ $ 0 $
$ 13 $ $ 0 $ $ 27 $ $ 81 $
$ 94 $ $ 135 $ $ 31 $ $ 45 $ $ 45 $ $ 0 $
$ 4 $ $ 18 $ $ 18 $ $ 81 $
$ 121 $ $ 108 $ $ 40 $ $ 36 $ $ 45 $ $ 0 $
$ 13 $ $ 9 $ $ 18 $ $ 81 $
$ 112 $ $ 126 $ $ 10 $ $ 15 $ $ 15 $ $ 81 $
$ 130 $ $ 117 $ $ 16 $ $ 12 $ $ 12 $ $ 81 $
$ 121 $ $ 135 $ $ 40 $ $ 45 $ $ 36 $ $ 0 $
$ 13 $ $ 18 $ $ 9 $ $ 81 $
$ 148 $ $ 108 $ $ 49 $ $ 36 $ $ 36 $ $ 0 $
$ 22 $ $ 9 $ $ 9 $ $ 81 $
Table 5.  $ q_{i,j} $ for $ (\varphi_0,\varphi_1) \in \mathcal{D}_6^+ $
$ \varphi_0 $ $ \varphi_1 $ $ q_{1,3} $ $ q_{0,2} $ $ q_{2,1} $
$ 121 $ $ 81 $ $ 13 $ $ 54 $ $ 54 $
$ 94 $ $ 135 $ $ 40 $ $ 54 $ $ 27 $
$ 121 $ $ 108 $ $ 31 $ $ 45 $ $ 45 $
$ 112 $ $ 126 $ $ 40 $ $ 45 $ $ 36 $
$ 130 $ $ 117 $ $ 40 $ $ 36 $ $ 45 $
$ 121 $ $ 135 $ $ 49 $ $ 36 $ $ 36 $
$ 148 $ $ 108 $ $ 40 $ $ 27 $ $ 54 $
$ \varphi_0 $ $ \varphi_1 $ $ q_{1,3} $ $ q_{0,2} $ $ q_{2,1} $
$ 121 $ $ 81 $ $ 13 $ $ 54 $ $ 54 $
$ 94 $ $ 135 $ $ 40 $ $ 54 $ $ 27 $
$ 121 $ $ 108 $ $ 31 $ $ 45 $ $ 45 $
$ 112 $ $ 126 $ $ 40 $ $ 45 $ $ 36 $
$ 130 $ $ 117 $ $ 40 $ $ 36 $ $ 45 $
$ 121 $ $ 135 $ $ 49 $ $ 36 $ $ 36 $
$ 148 $ $ 108 $ $ 40 $ $ 27 $ $ 54 $
Table 6.  $r_{i,j}$ for $(\varphi_0,\varphi_1) \in \mathcal{D}_6^+$
$\varphi_0$ $\varphi_1$ $r_{1,0}$ $r_{0,2}$ $r_{2,1}$
121 81 67 27 27
94 135 40 54 27
121 108 49 36 36
112 126 40 45 36
130 117 40 36 45
121 135 31 45 45
148 108 40 27 54
$\varphi_0$ $\varphi_1$ $r_{1,0}$ $r_{0,2}$ $r_{2,1}$
121 81 67 27 27
94 135 40 54 27
121 108 49 36 36
112 126 40 45 36
130 117 40 36 45
121 135 31 45 45
148 108 40 27 54
Table 7.  The spectra of some ternary linear codes of dimension 4 [31]
parameters possible spectra
$ [4,4,1]_3 $ $ (a_0, a_1, a_2, a_3)=(8,16,12,4) $
$ [5,4,2]_3 $ $ (a_0, a_1, a_2, a_3)=(5,15,10,10) $
$ [7,4,3]_3 $ $ (a_0, a_1, a_2, a_3, a_4)=(3,8,9,15,5) $
$ (a_1, a_2, a_3, a_4)=(14,9,9,8) $
$ (a_0, a_1, a_2, a_3, a_4)=(2,9,12,10,7) $
$ (a_0, a_1, a_2, a_3, a_4)=(4,4,15,11,6) $
$ [8,4,4]_3 $ $ (a_0, a_1, a_2, a_3, a_4)=(3,4,10,12,11) $
$ (a_0, a_1, a_2, a_3, a_4)=(2,8,4,16,10) $
$ (a_0, a_2, a_3, a_4)=(4,16,8,12) $
$ [9,4,5]_3 $ $ (a_0,a_1,a_3, a_4)=(1,9,12,18) $
$ [10,4,6]_3 $ $ (a_1, a_4)=(10,30) $
$ [14,4,8]_3 $ $ (a_1, a_2, a_3, a_4, a_5, a_6)=(1,4,4,8,9,14) $
$ (a_1, a_2, a_4, a_5, a_6)=(2,4,10,12,12) $
$ (a_1, a_2, a_3, a_4, a_5, a_6)=(2,2,5,7,11,13) $
$ (a_1, a_2, a_3, a_4, a_5, a_6)=(3,1,2,12,9,13) $
$ (a_1, a_2, a_3, a_4, a_5, a_6)=(3,3,3,6,10,15) $
$ (a_0, a_2, a_3, a_4, a_5, a_6)=(1,3,4,9,10,13) $
$ (a_0, a_2, a_3, a_5, a_6)=(1,3,10,10,16) $
$ (a_2, a_3, a_5, a_6)=(3,12,10,15) $
$ (a_0, a_2, a_4, a_5, a_6)=(1,4,15,6,14) $
$ (a_0,, a_3, a_5, a_6)=(1,13,13,13) $
$ [15,4,9]_3 $ $ (a_0, a_3, a_6)=(1,13,26) $
$ (a_3, a_6)=(15,25) $
$ [19,4,12]_3 $ $ (a_{1},a_{4},a_{7})=(1,9,30) $
$ [25,4,16]_3 $ $ (a_{0},a_{7},a_{8},a_{9})=(1,4,18,17) $
$ [26,4,17]_3 $ $ (a_0,a_8,a_{9}) = (1,13,26) $
$ [27,4,18]_3 $ $ (a_0,a_{9}) = (1,39) $
$ [31,4,20]_3 $ $ (a_4,a_9,a_{10},a_{11})=(1,9,12,18) $
$ (a_{7},a_{8},a_{10},a_{11})=(2,6,11,21) $
$ [32,4,21]_3 $ $ (a_{8},a_{11})=(8,32) $
parameters possible spectra
$ [4,4,1]_3 $ $ (a_0, a_1, a_2, a_3)=(8,16,12,4) $
$ [5,4,2]_3 $ $ (a_0, a_1, a_2, a_3)=(5,15,10,10) $
$ [7,4,3]_3 $ $ (a_0, a_1, a_2, a_3, a_4)=(3,8,9,15,5) $
$ (a_1, a_2, a_3, a_4)=(14,9,9,8) $
$ (a_0, a_1, a_2, a_3, a_4)=(2,9,12,10,7) $
$ (a_0, a_1, a_2, a_3, a_4)=(4,4,15,11,6) $
$ [8,4,4]_3 $ $ (a_0, a_1, a_2, a_3, a_4)=(3,4,10,12,11) $
$ (a_0, a_1, a_2, a_3, a_4)=(2,8,4,16,10) $
$ (a_0, a_2, a_3, a_4)=(4,16,8,12) $
$ [9,4,5]_3 $ $ (a_0,a_1,a_3, a_4)=(1,9,12,18) $
$ [10,4,6]_3 $ $ (a_1, a_4)=(10,30) $
$ [14,4,8]_3 $ $ (a_1, a_2, a_3, a_4, a_5, a_6)=(1,4,4,8,9,14) $
$ (a_1, a_2, a_4, a_5, a_6)=(2,4,10,12,12) $
$ (a_1, a_2, a_3, a_4, a_5, a_6)=(2,2,5,7,11,13) $
$ (a_1, a_2, a_3, a_4, a_5, a_6)=(3,1,2,12,9,13) $
$ (a_1, a_2, a_3, a_4, a_5, a_6)=(3,3,3,6,10,15) $
$ (a_0, a_2, a_3, a_4, a_5, a_6)=(1,3,4,9,10,13) $
$ (a_0, a_2, a_3, a_5, a_6)=(1,3,10,10,16) $
$ (a_2, a_3, a_5, a_6)=(3,12,10,15) $
$ (a_0, a_2, a_4, a_5, a_6)=(1,4,15,6,14) $
$ (a_0,, a_3, a_5, a_6)=(1,13,13,13) $
$ [15,4,9]_3 $ $ (a_0, a_3, a_6)=(1,13,26) $
$ (a_3, a_6)=(15,25) $
$ [19,4,12]_3 $ $ (a_{1},a_{4},a_{7})=(1,9,30) $
$ [25,4,16]_3 $ $ (a_{0},a_{7},a_{8},a_{9})=(1,4,18,17) $
$ [26,4,17]_3 $ $ (a_0,a_8,a_{9}) = (1,13,26) $
$ [27,4,18]_3 $ $ (a_0,a_{9}) = (1,39) $
$ [31,4,20]_3 $ $ (a_4,a_9,a_{10},a_{11})=(1,9,12,18) $
$ (a_{7},a_{8},a_{10},a_{11})=(2,6,11,21) $
$ [32,4,21]_3 $ $ (a_{8},a_{11})=(8,32) $
Table 8.  The spectra of some ternary linear codes of dimension 5
parameters possible spectra reference
$ [11,5,6]_3 $ $ (a_{2},a_{5})=(55,66) $ [31]
$ [20,5,12]_3 $ $ (a_{2},a_{5},a_{8})=(10,36,75) $ [31]
$ [25,5,15]_3 $ $ (a_4,a_7,a_{10})=(15,40,66) $ [2]
$ (a_1,a_4,a_7,a_{10})=(1,12,43,65) $
$ [29,5,18]_3 $ $ (a_{2},a_{5},a_{8},a_{11})=(1,18,18,84) $ [2]
$ [55,5,36]_3 $ $ (a_{10},a_{19}) = (11,110) $ [7]
$ [68,5,44]_3 $ $ (a_{14},a_{15},a_{23},a_{24}) = (1,15,39,65) $ [27]
$ (a_{14},a_{15},a_{23},a_{24}) = (4,12,36,69) $
$ [69,5,45]_3 $ $ (a_{15},a_{24}) = (16,105) $ [30]
$ [79,5,52]_3 $ $ (a_0,a_{25},a_{26},a_{27})=(1,13,54,53) $ [5]
$ [80,5,53]_3 $ $ (a_0,a_{26},a_{27})=(1,40,80) $ [5]
$ [81,5,54]_3 $ $ (a_0,a_{27})=(1,120) $ [5]
$ [87,5,57]_3 $ $ (a_9,a_{24},a_{27},a_{30})=(1,1,41,78) $ [6]
$ [90,5,59]_3 $ $ (a_{10},a_{27},a_{28},a_{30},a_{31}) = (1,10,20,30,60) $ [26]
$ (a_{9},a_{27},a_{28},a_{30},a_{31}) = (1,3,27,36,54) $
$ [91,5,60]_3 $ $ (a_{10},a_{28},a_{31}) = (1,30,90) $ [32]
parameters possible spectra reference
$ [11,5,6]_3 $ $ (a_{2},a_{5})=(55,66) $ [31]
$ [20,5,12]_3 $ $ (a_{2},a_{5},a_{8})=(10,36,75) $ [31]
$ [25,5,15]_3 $ $ (a_4,a_7,a_{10})=(15,40,66) $ [2]
$ (a_1,a_4,a_7,a_{10})=(1,12,43,65) $
$ [29,5,18]_3 $ $ (a_{2},a_{5},a_{8},a_{11})=(1,18,18,84) $ [2]
$ [55,5,36]_3 $ $ (a_{10},a_{19}) = (11,110) $ [7]
$ [68,5,44]_3 $ $ (a_{14},a_{15},a_{23},a_{24}) = (1,15,39,65) $ [27]
$ (a_{14},a_{15},a_{23},a_{24}) = (4,12,36,69) $
$ [69,5,45]_3 $ $ (a_{15},a_{24}) = (16,105) $ [30]
$ [79,5,52]_3 $ $ (a_0,a_{25},a_{26},a_{27})=(1,13,54,53) $ [5]
$ [80,5,53]_3 $ $ (a_0,a_{26},a_{27})=(1,40,80) $ [5]
$ [81,5,54]_3 $ $ (a_0,a_{27})=(1,120) $ [5]
$ [87,5,57]_3 $ $ (a_9,a_{24},a_{27},a_{30})=(1,1,41,78) $ [6]
$ [90,5,59]_3 $ $ (a_{10},a_{27},a_{28},a_{30},a_{31}) = (1,10,20,30,60) $ [26]
$ (a_{9},a_{27},a_{28},a_{30},a_{31}) = (1,3,27,36,54) $
$ [91,5,60]_3 $ $ (a_{10},a_{28},a_{31}) = (1,30,90) $ [32]
Table 9.  All solutions of (15) with $ w = 8 $
$ t $ solution line in $ \Sigma^* $ # $ L_e $
0 $ (c_{13},c_{22},c_{24})=(1,1,1) $ $ (2,1) $ $ x_1 $ 56
$ (c_{14},c_{22},c_{23})=(1,1,1) $ $ (1,3) $ $ x_2 $ 46
$ (c_{16},c_{19},c_{24})=(1,1,1) $ $ (2,1) $ $ x_3 $ 38
$ (c_{17},c_{18},c_{24})=(1,1,1) $ $ (0,2) $ $ x_4 $ 36
$ (c_{16},c_{20},c_{23})=(1,1,1) $ $ (1,3) $ $ x_5 $ 34
$ (c_{17},c_{19},c_{23})=(1,1,1) $ $ (1,3) $ $ x_6 $ 31
$ (c_{18},c_{23})=(2,1) $ $ (0,2) $ $ x_7 $ 30
$ (c_{17},c_{20},c_{22})=(1,1,1) $ $ (1,3) $ $ x_8 $ 28
$ (c_{18},c_{19},c_{22})=(1,1,1) $ $ (2,1) $ $ x_9 $ 26
$ (c_{19},c_{20})=(1,2) $ $ (1,3) $ $ x_{10} $ 22
1 $ (c_{16},c_{23})=(1,2) $ $ (1,3) $ $ x_{11} $ 28
$ (c_{17},c_{22},c_{23})=(1,1,1) $ $ (1,3) $ $ x_{12} $ 22
$ (c_{18},c_{22})=(1,2) $ $ (2,1) $ $ x_{13} $ 17
$ (c_{19},c_{20},c_{23})=(1,1,1) $ $ (1,3) $ $ x_{14} $ 16
$ (c_{20},c_{22})=(2,1) $ $ (1,3) $ $ x_{15} $ 13
2 $ (c_{19},c_{23})=(1,2) $ $ (1,3) $ $ x_{16} $ 10
$ (c_{20},c_{22},c_{23})=(1,1,1) $ $ (1,3) $ $ x_{17} $ 7
3 $ (c_{20},c_{24})=(1,2) $ $ (0,2) $ $ x_{18} $ 6
$ (c_{22},c_{24})=(2,1) $ $ (2,1) $ $ x_{19} $ 2
$ (c_{22},c_{23})=(1,2) $ $ (1,3) $ $ x_{20} $ 1
4 $ (c_{23},c_{24})=(1,2) $ $ (0,2) $ $ x_{21} $ 0
$ t $ solution line in $ \Sigma^* $ # $ L_e $
0 $ (c_{13},c_{22},c_{24})=(1,1,1) $ $ (2,1) $ $ x_1 $ 56
$ (c_{14},c_{22},c_{23})=(1,1,1) $ $ (1,3) $ $ x_2 $ 46
$ (c_{16},c_{19},c_{24})=(1,1,1) $ $ (2,1) $ $ x_3 $ 38
$ (c_{17},c_{18},c_{24})=(1,1,1) $ $ (0,2) $ $ x_4 $ 36
$ (c_{16},c_{20},c_{23})=(1,1,1) $ $ (1,3) $ $ x_5 $ 34
$ (c_{17},c_{19},c_{23})=(1,1,1) $ $ (1,3) $ $ x_6 $ 31
$ (c_{18},c_{23})=(2,1) $ $ (0,2) $ $ x_7 $ 30
$ (c_{17},c_{20},c_{22})=(1,1,1) $ $ (1,3) $ $ x_8 $ 28
$ (c_{18},c_{19},c_{22})=(1,1,1) $ $ (2,1) $ $ x_9 $ 26
$ (c_{19},c_{20})=(1,2) $ $ (1,3) $ $ x_{10} $ 22
1 $ (c_{16},c_{23})=(1,2) $ $ (1,3) $ $ x_{11} $ 28
$ (c_{17},c_{22},c_{23})=(1,1,1) $ $ (1,3) $ $ x_{12} $ 22
$ (c_{18},c_{22})=(1,2) $ $ (2,1) $ $ x_{13} $ 17
$ (c_{19},c_{20},c_{23})=(1,1,1) $ $ (1,3) $ $ x_{14} $ 16
$ (c_{20},c_{22})=(2,1) $ $ (1,3) $ $ x_{15} $ 13
2 $ (c_{19},c_{23})=(1,2) $ $ (1,3) $ $ x_{16} $ 10
$ (c_{20},c_{22},c_{23})=(1,1,1) $ $ (1,3) $ $ x_{17} $ 7
3 $ (c_{20},c_{24})=(1,2) $ $ (0,2) $ $ x_{18} $ 6
$ (c_{22},c_{24})=(2,1) $ $ (2,1) $ $ x_{19} $ 2
$ (c_{22},c_{23})=(1,2) $ $ (1,3) $ $ x_{20} $ 1
4 $ (c_{23},c_{24})=(1,2) $ $ (0,2) $ $ x_{21} $ 0
Table 10.  All solutions of (25) with $ w = 49 $
$ t $ solution line in $ \Sigma^* $ # $ L_e $
8 $ (c_{77},c_{89},c_{92})=(1,1,1) $ $ (1,3) $ $ x_1 $ 198
$ c_{86}=3 $ $ (1,3) $ $ x_2 $ 135
9 $ (c_{77},c_{92})=(1,2) $ $ (1,3) $ $ x_3 $ 183
$ (c_{86},c_{89})=(2,1) $ $ (1,3) $ $ x_4 $ 111
$ c_{87}=3 $ $ (1,0) $ $ x_5 $ 108
13 $ (c_{85},c_{94})=(1,2) $ $ (4,0) $ $ x_6 $ 57
$ (c_{88},c_{91},c_{94})=(1,1,1) $ $ (4,0) $ $ x_7 $ 39
$ c_{91}=3 $ $ (4,0) $ $ x_8 $ 30
16 $ c_{94}=3 $ $ (4,0) $ $ 36 $ 3
17 $ c_{95}=3 $ $ (1,3) $ $ 26 $ 0
18 $ c_{96}=3 $ $ (1,0) $ $ 44 $ 0
$ t $ solution line in $ \Sigma^* $ # $ L_e $
8 $ (c_{77},c_{89},c_{92})=(1,1,1) $ $ (1,3) $ $ x_1 $ 198
$ c_{86}=3 $ $ (1,3) $ $ x_2 $ 135
9 $ (c_{77},c_{92})=(1,2) $ $ (1,3) $ $ x_3 $ 183
$ (c_{86},c_{89})=(2,1) $ $ (1,3) $ $ x_4 $ 111
$ c_{87}=3 $ $ (1,0) $ $ x_5 $ 108
13 $ (c_{85},c_{94})=(1,2) $ $ (4,0) $ $ x_6 $ 57
$ (c_{88},c_{91},c_{94})=(1,1,1) $ $ (4,0) $ $ x_7 $ 39
$ c_{91}=3 $ $ (4,0) $ $ x_8 $ 30
16 $ c_{94}=3 $ $ (4,0) $ $ 36 $ 3
17 $ c_{95}=3 $ $ (1,3) $ $ 26 $ 0
18 $ c_{96}=3 $ $ (1,0) $ $ 44 $ 0
Table 11.  Values and bounds for $ n_3(6,d) $ for $ d \leq 351 $
$ d $ $ g_3(6,d) $ $ n_3(6,d) $ $ d $ $ g_3(6,d) $ $ n_3(6,d) $ $ d $ $ g_3(6,d) $ $ n_3(6,d) $
1 6 6 61 94 96 121 184 185
2 7 7 62 95 97 122 185 186
3 8 9 63 96 98 123 186 187
4 10 10 64 99 100-101 124 188 189
5 11 11 65 100 101-102 125 189 190
6 12 12 66 101 103 126 190 191
7 14 15 67 103 105 127 193 194-195
8 15 17 68 104 106 128 194 195-196
9 16 18 69 105 107 129 195 196-197
10 19 20 70 107 109 130 197 199
11 20 21 71 108 110 131 198 200
12 21 22 72 109 111 132 199 201
13 23 24 73 112 114 133 201 203
14 24 25 74 113 115 134 202 204
15 25 26 75 114 116 135 203 205
16 27 29 76 116 118 136 207 208-209
17 28 30 77 117 119 137 208 209-210
18 29 31 78 118 120 138 209 210-211
19 32 33-34 79 120 122 139 211 212-213
20 33 34-35 80 121 123 140 212 213-214
21 34 36 81 122 124 141 213 214-215
22 36 38 82 127 127-128 142 215 216-217
23 37 39 83 128 128-129 143 216 217-218
24 38 40 84 129 129-130 144 217 218-219
25 40 42 85 131 131-132 145 220 221-222
26 41 43 86 132 133 146 221 222-223
27 42 44 87 133 134 147 222 223-224
28 46 46-47 88 135 136 148 224 225-226
29 47 48 89 136 137 149 225 227
30 48 49 90 137 138 150 226 228
31 50 51 91 140 140-142 151 228 230
32 51 52 92 141 141-143 152 229 231
33 52 53 93 142 143-144 153 230 232
34 54 54 94 144 145-146 154 233 234
35 55 55 95 145 146-147 155 234 235
36 56 56 96 146 147-148 156 235 236
$ d $ $ g_3(6,d) $ $ n_3(6,d) $ $ d $ $ g_3(6,d) $ $ n_3(6,d) $ $ d $ $ g_3(6,d) $ $ n_3(6,d) $
1 6 6 61 94 96 121 184 185
2 7 7 62 95 97 122 185 186
3 8 9 63 96 98 123 186 187
4 10 10 64 99 100-101 124 188 189
5 11 11 65 100 101-102 125 189 190
6 12 12 66 101 103 126 190 191
7 14 15 67 103 105 127 193 194-195
8 15 17 68 104 106 128 194 195-196
9 16 18 69 105 107 129 195 196-197
10 19 20 70 107 109 130 197 199
11 20 21 71 108 110 131 198 200
12 21 22 72 109 111 132 199 201
13 23 24 73 112 114 133 201 203
14 24 25 74 113 115 134 202 204
15 25 26 75 114 116 135 203 205
16 27 29 76 116 118 136 207 208-209
17 28 30 77 117 119 137 208 209-210
18 29 31 78 118 120 138 209 210-211
19 32 33-34 79 120 122 139 211 212-213
20 33 34-35 80 121 123 140 212 213-214
21 34 36 81 122 124 141 213 214-215
22 36 38 82 127 127-128 142 215 216-217
23 37 39 83 128 128-129 143 216 217-218
24 38 40 84 129 129-130 144 217 218-219
25 40 42 85 131 131-132 145 220 221-222
26 41 43 86 132 133 146 221 222-223
27 42 44 87 133 134 147 222 223-224
28 46 46-47 88 135 136 148 224 225-226
29 47 48 89 136 137 149 225 227
30 48 49 90 137 138 150 226 228
31 50 51 91 140 140-142 151 228 230
32 51 52 92 141 141-143 152 229 231
33 52 53 93 142 143-144 153 230 232
34 54 54 94 144 145-146 154 233 234
35 55 55 95 145 146-147 155 234 235
36 56 56 96 146 147-148 156 235 236
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