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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 |
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.
References:
[1] |
I. G. Bouyukliev,
What is Q-Extension?, Serdica J. Computing, 1 (2007), 115-130.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
R. Hill,
Caps and codes, Discrete Math., 22 (1978), 111-137.
doi: 10.1016/0012-365X(78)90120-6. |
[8] |
R. Hill,
Optimal linear codes, Cryptography and Coding, 33 (1992), 75-104.
|
[9] |
R. Hill,
An extension theorem for linear codes, Des. Codes Cryptogr., 17 (1999), 151-157.
doi: 10.1023/A:1008319024396. |
[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. |
[11] |
R. Hill and D. E. Newton,
Optimal ternary linear codes, Des. Codes Cryptogr., 2 (1992), 137-157.
doi: 10.1007/BF00124893. |
[12] |
W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511807077.![]() ![]() ![]() |
[13] |
C. M. Jones, Optimal Ternary Linear Codes, PhD thesis, University of Salford, 2000. |
[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. |
[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. |
[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. |
[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. |
[18] |
I. Landgev, T. 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. |
[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. |
[20] |
T. Maruta,
On the achievement of the Griesmer bound, Des. Codes Cryptogr., 12 (1997), 83-87.
doi: 10.1023/A:1008250010928. |
[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. |
[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. |
[23] |
T. Maruta,
Extendability of ternary linear codes, Des. Codes Cryptogr., 35 (2005), 175-190.
doi: 10.1007/s10623-005-6400-7. |
[24] |
T. Maruta, Griesmer bound for linear codes over finite fields, Available from: http://mars39.lomo.jp/opu/griesmer.htm. |
[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. |
[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. |
[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. |
[28] |
T. Sawashima and T. Maruta,
Nonexistence of some ternary linear codes, Discrete Math., 344 (2021), 112572.
doi: 10.1016/j.disc.2021.112572. |
[29] |
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. |
[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. |
[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. |
[32] |
Y. Yoshida and T. Maruta, Ternary linear codes and quadrics, Electronic J. Combin., 16 (2009), #R9, 21pp.
doi: 10.37236/98. |
show all references
References:
[1] |
I. G. Bouyukliev,
What is Q-Extension?, Serdica J. Computing, 1 (2007), 115-130.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
R. Hill,
Caps and codes, Discrete Math., 22 (1978), 111-137.
doi: 10.1016/0012-365X(78)90120-6. |
[8] |
R. Hill,
Optimal linear codes, Cryptography and Coding, 33 (1992), 75-104.
|
[9] |
R. Hill,
An extension theorem for linear codes, Des. Codes Cryptogr., 17 (1999), 151-157.
doi: 10.1023/A:1008319024396. |
[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. |
[11] |
R. Hill and D. E. Newton,
Optimal ternary linear codes, Des. Codes Cryptogr., 2 (1992), 137-157.
doi: 10.1007/BF00124893. |
[12] |
W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511807077.![]() ![]() ![]() |
[13] |
C. M. Jones, Optimal Ternary Linear Codes, PhD thesis, University of Salford, 2000. |
[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. |
[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. |
[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. |
[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. |
[18] |
I. Landgev, T. 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. |
[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. |
[20] |
T. Maruta,
On the achievement of the Griesmer bound, Des. Codes Cryptogr., 12 (1997), 83-87.
doi: 10.1023/A:1008250010928. |
[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. |
[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. |
[23] |
T. Maruta,
Extendability of ternary linear codes, Des. Codes Cryptogr., 35 (2005), 175-190.
doi: 10.1007/s10623-005-6400-7. |
[24] |
T. Maruta, Griesmer bound for linear codes over finite fields, Available from: http://mars39.lomo.jp/opu/griesmer.htm. |
[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. |
[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. |
[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. |
[28] |
T. Sawashima and T. Maruta,
Nonexistence of some ternary linear codes, Discrete Math., 344 (2021), 112572.
doi: 10.1016/j.disc.2021.112572. |
[29] |
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. |
[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. |
[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. |
[32] |
Y. Yoshida and T. Maruta, Ternary linear codes and quadrics, Electronic J. Combin., 16 (2009), #R9, 21pp.
doi: 10.37236/98. |
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 |
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 |
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 |
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 |
parameters | possible spectra |
parameters | possible spectra |
parameters | possible spectra | reference |
[31] | ||
[31] | ||
[2] | ||
[2] | ||
[7] | ||
[27] | ||
[30] | ||
[5] | ||
[5] | ||
[5] | ||
[6] | ||
[26] | ||
[32] |
parameters | possible spectra | reference |
[31] | ||
[31] | ||
[2] | ||
[2] | ||
[7] | ||
[27] | ||
[30] | ||
[5] | ||
[5] | ||
[5] | ||
[6] | ||
[26] | ||
[32] |
solution | line in |
# | ||
0 | 56 | |||
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38 | ||||
36 | ||||
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31 | ||||
30 | ||||
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1 | 28 | |||
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2 | 10 | |||
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3 | 6 | |||
2 | ||||
1 | ||||
4 | 0 |
solution | line in |
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0 | 56 | |||
46 | ||||
38 | ||||
36 | ||||
34 | ||||
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30 | ||||
28 | ||||
26 | ||||
22 | ||||
1 | 28 | |||
22 | ||||
17 | ||||
16 | ||||
13 | ||||
2 | 10 | |||
7 | ||||
3 | 6 | |||
2 | ||||
1 | ||||
4 | 0 |
solution | line in |
# | ||
8 | 198 | |||
135 | ||||
9 | 183 | |||
111 | ||||
108 | ||||
13 | 57 | |||
39 | ||||
30 | ||||
16 | 3 | |||
17 | 0 | |||
18 | 0 |
solution | line in |
# | ||
8 | 198 | |||
135 | ||||
9 | 183 | |||
111 | ||||
108 | ||||
13 | 57 | |||
39 | ||||
30 | ||||
16 | 3 | |||
17 | 0 | |||
18 | 0 |
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 |
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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Ali Tebbi, Terence Chan, Chi Wan Sung. Linear programming bounds for distributed storage codes. Advances in Mathematics of Communications, 2020, 14 (2) : 333-357. doi: 10.3934/amc.2020024 |
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