# American Institute of Mathematical Sciences

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

Received  September 2019 Revised  January 2020 Published  April 2020

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.

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.  Google Scholar [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.  Google Scholar [3] I. G. Bouyukliev, What is $Q$-extension?, Serdica Journal of Computing, 1 (2007), 115-130.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [8] S. Kurz, Lincode - computer classification of linear codes, arXiv: 1912.09357. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] H. N. Ward, Divisible codes - a survey, Serdica Mathematical Journal, 27 (2001), 263-278.   Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [3] I. G. Bouyukliev, What is $Q$-extension?, Serdica Journal of Computing, 1 (2007), 115-130.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [8] S. Kurz, Lincode - computer classification of linear codes, arXiv: 1912.09357. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] H. N. Ward, Divisible codes - a survey, Serdica Mathematical Journal, 27 (2001), 263-278.   Google Scholar
Number of $[n,k,\{20,24,28,32\}]_2$ codes
 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
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
Number of $[84,8,\{40,48,56\}]_2$ codes per $A_{56}$
 $A_{56}$ 3 4 5 6 7 8 9 25773 48792 26091 5198 450 17 1
 $A_{56}$ 3 4 5 6 7 8 9 25773 48792 26091 5198 450 17 1
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