Table 1.
Number of $ [n,k,\{20,24,28,32\}]_2 $ codes
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August
2021, 15(3): 415-422.
doi: 10.3934/amc.2020074
The $[46, 9, 20]_2$ code is unique
Mathematisches Institut, Universität Bayreuth, D-95440 Bayreuth, Germany |
The minimum distance of all binary linear codes with dimension at most eight is known. The smallest open case for dimension nine is length $ n = 46 $ with known bounds $ 19\le d\le 20 $. Here we present a $ [46,9,20]_2 $ code and show its uniqueness. Interestingly enough, this unique optimal code is asymmetric, i.e., it has a trivial automorphism group. Additionally, we show the non-existence of $ [47,10,20]_2 $ and $ [85,9,40]_2 $ codes.
Keywords:
Binary linear codes,
optimal codes,
divisible codes,
non-existence of codes,
computer search.
Mathematics Subject Classification:
Primary: 94B65; Secondary: 94B05.
Citation:
Sascha Kurz. The $[46, 9, 20]_2$ code is unique. Advances in Mathematics of Communications,
2021, 15
(3)
: 415-422.
doi: 10.3934/amc.2020074
![]()
References:
| [1] |
L. D. Baumert and R. J. McEliece,
A note on the Griesmer bound, IEEE Transactions on Information Theory, IT-19 (1973), 134-135.
doi: 10.1109/tit.1973.1054939. |
| [2] |
I. Bouyukliev, D. B. Jaffe and V. Vavrek,
The smallest length of eight-dimensional binary linear codes with prescribed minimum distance, IEEE Transactions on Information Theory, 46 (2000), 1539-1544.
doi: 10.1109/18.850690. |
| [3] |
I. G. Bouyukliev,
What is $Q$-extension?, Serdica Journal of Computing, 1 (2007), 115-130.
|
| [4] |
I. Bouyukliev and D. B. Jaffe,
Optimal binary linear codes of dimension at most seven, Discrete Mathematics, 226 (2001), 51-70.
doi: 10.1016/S0012-365X(00)00125-4. |
| [5] |
S. Dodunekov, S. Guritman and J. Simonis,
Some new results on the minimum length of binary linear codes of dimension nine, IEEE Transactions on Information Theory, 45 (1999), 2543-2546.
doi: 10.1109/18.796403. |
| [6] |
M. Grassl, Bounds on the minimum distance of linear codes and quantum codes, Online available at: http://www.codetables.de, (2007), Accessed on 2019-04-04. |
| [7] |
J. H. Griesmer,
A bound for error-correcting codes, IBM Journal of Research and Development, 4 (1960), 532-542.
doi: 10.1147/rd.45.0532. |
| [8] |
S. Kurz, Lincode - computer classification of linear codes, arXiv: 1912.09357. |
| [9] |
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. II, North-Holland Mathematical Library, Vol. 16. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. |
| [10] |
J. Simonis,
Restrictions on the weight distribution of binary linear codes imposed by the structure of reed-muller codes, IEEE transactions on Information Theory, 40 (1994), 194-196.
doi: 10.1109/18.272480. |
| [11] |
J. Simonis,
The $[23, 14, 5]$ Wagner code is unique, Discrete Mathematics, 213 (2000), 269-282.
doi: 10.1016/S0012-365X(99)00187-9. |
| [12] |
H. C. A. van Tilborg,
The smallest length of binary $7$-dimensional linear codes with prescribed minimum distance, Discrete Mathematics, 33 (1981), 197-207.
doi: 10.1016/0012-365X(81)90166-7. |
| [13] |
H. N. Ward,
Divisible codes - a survey, Serdica Mathematical Journal, 27 (2001), 263-278.
|
show all references
References:
| [1] |
L. D. Baumert and R. J. McEliece,
A note on the Griesmer bound, IEEE Transactions on Information Theory, IT-19 (1973), 134-135.
doi: 10.1109/tit.1973.1054939. |
| [2] |
I. Bouyukliev, D. B. Jaffe and V. Vavrek,
The smallest length of eight-dimensional binary linear codes with prescribed minimum distance, IEEE Transactions on Information Theory, 46 (2000), 1539-1544.
doi: 10.1109/18.850690. |
| [3] |
I. G. Bouyukliev,
What is $Q$-extension?, Serdica Journal of Computing, 1 (2007), 115-130.
|
| [4] |
I. Bouyukliev and D. B. Jaffe,
Optimal binary linear codes of dimension at most seven, Discrete Mathematics, 226 (2001), 51-70.
doi: 10.1016/S0012-365X(00)00125-4. |
| [5] |
S. Dodunekov, S. Guritman and J. Simonis,
Some new results on the minimum length of binary linear codes of dimension nine, IEEE Transactions on Information Theory, 45 (1999), 2543-2546.
doi: 10.1109/18.796403. |
| [6] |
M. Grassl, Bounds on the minimum distance of linear codes and quantum codes, Online available at: http://www.codetables.de, (2007), Accessed on 2019-04-04. |
| [7] |
J. H. Griesmer,
A bound for error-correcting codes, IBM Journal of Research and Development, 4 (1960), 532-542.
doi: 10.1147/rd.45.0532. |
| [8] |
S. Kurz, Lincode - computer classification of linear codes, arXiv: 1912.09357. |
| [9] |
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. II, North-Holland Mathematical Library, Vol. 16. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. |
| [10] |
J. Simonis,
Restrictions on the weight distribution of binary linear codes imposed by the structure of reed-muller codes, IEEE transactions on Information Theory, 40 (1994), 194-196.
doi: 10.1109/18.272480. |
| [11] |
J. Simonis,
The $[23, 14, 5]$ Wagner code is unique, Discrete Mathematics, 213 (2000), 269-282.
doi: 10.1016/S0012-365X(99)00187-9. |
| [12] |
H. C. A. van Tilborg,
The smallest length of binary $7$-dimensional linear codes with prescribed minimum distance, Discrete Mathematics, 33 (1981), 197-207.
doi: 10.1016/0012-365X(81)90166-7. |
| [13] |
H. N. Ward,
Divisible codes - a survey, Serdica Mathematical Journal, 27 (2001), 263-278.
|
| k/n | 20 | 24 | 28 | 30 | 32 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 |
| 1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | ||||||
| 2 | 1 | 1 | 2 | 0 | 3 | 0 | 3 | 0 | ||||||||
| 3 | 1 | 1 | 2 | 4 | 6 | 9 | ||||||||||
| 4 | 1 | 4 | 13 | 26 | ||||||||||||
| 5 | 3 | 15 | 163 | |||||||||||||
| 6 | 24 | 3649 | ||||||||||||||
| 7 | 5 | 337794 |
| k/n | 20 | 24 | 28 | 30 | 32 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 |
| 1 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | ||||||
| 2 | 1 | 1 | 2 | 0 | 3 | 0 | 3 | 0 | ||||||||
| 3 | 1 | 1 | 2 | 4 | 6 | 9 | ||||||||||
| 4 | 1 | 4 | 13 | 26 | ||||||||||||
| 5 | 3 | 15 | 163 | |||||||||||||
| 6 | 24 | 3649 | ||||||||||||||
| 7 | 5 | 337794 |
Table 2.
Number of $ [n,k,\{40,48,56\}]_2 $ codes
| k/n | 40 | 48 | 56 | 60 | 64 | 68 | 70 | 72 | 74 | 75 | 76 | 77 | 78 | 79 | 80 | 81 | 82 | 83 |
| 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||||||
| 2 | 1 | 1 | 2 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | ||||||||
| 3 | 1 | 1 | 2 | 0 | 3 | 0 | 5 | 0 | ||||||||||
| 4 | 1 | 1 | 2 | 3 | 6 | 10 | ||||||||||||
| 5 | 1 | 3 | 11 | 16 | ||||||||||||||
| 6 | 2 | 8 | 106 | |||||||||||||||
| 7 | 7 | 5613 |
| k/n | 40 | 48 | 56 | 60 | 64 | 68 | 70 | 72 | 74 | 75 | 76 | 77 | 78 | 79 | 80 | 81 | 82 | 83 |
| 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||||||
| 2 | 1 | 1 | 2 | 0 | 2 | 0 | 0 | 2 | 0 | 0 | ||||||||
| 3 | 1 | 1 | 2 | 0 | 3 | 0 | 5 | 0 | ||||||||||
| 4 | 1 | 1 | 2 | 3 | 6 | 10 | ||||||||||||
| 5 | 1 | 3 | 11 | 16 | ||||||||||||||
| 6 | 2 | 8 | 106 | |||||||||||||||
| 7 | 7 | 5613 |
| 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
| 25773 | 48792 | 26091 | 5198 | 450 | 17 | 1 |
| 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
| 25773 | 48792 | 26091 | 5198 | 450 | 17 | 1 |
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