# 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  August 2021 Early access  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. [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.
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
 [1] Michael Braun. On lattices, binary codes, and network codes. Advances in Mathematics of Communications, 2011, 5 (2) : 225-232. doi: 10.3934/amc.2011.5.225 [2] Yansheng Wu, Jong Yoon Hyun, Qin Yue. Optimal binary linear codes from posets of the disjoint union of two chains. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022042 [3] Axel Kohnert, Johannes Zwanzger. New linear codes with prescribed group of automorphisms found by heuristic search. Advances in Mathematics of Communications, 2009, 3 (2) : 157-166. doi: 10.3934/amc.2009.3.157 [4] Joaquim Borges, Ivan Yu. Mogilnykh, Josep Rifà, Faina I. Solov'eva. Structural properties of binary propelinear codes. Advances in Mathematics of Communications, 2012, 6 (3) : 329-346. doi: 10.3934/amc.2012.6.329 [5] 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 [6] Tsonka Baicheva, Iliya Bouyukliev. On the least covering radius of binary linear codes of dimension 6. Advances in Mathematics of Communications, 2010, 4 (3) : 399-404. doi: 10.3934/amc.2010.4.399 [7] Olof Heden, Fabio Pasticci, Thomas Westerbäck. On the existence of extended perfect binary codes with trivial symmetry group. Advances in Mathematics of Communications, 2009, 3 (3) : 295-309. doi: 10.3934/amc.2009.3.295 [8] Hans-Joachim Kroll, Sayed-Ghahreman Taherian, Rita Vincenti. Optimal antiblocking systems of information sets for the binary codes related to triangular graphs. Advances in Mathematics of Communications, 2022, 16 (1) : 169-183. doi: 10.3934/amc.2020107 [9] Yael Ben-Haim, Simon Litsyn. A new upper bound on the rate of non-binary codes. Advances in Mathematics of Communications, 2007, 1 (1) : 83-92. doi: 10.3934/amc.2007.1.83 [10] Olof Heden, Martin Hessler. On linear equivalence and Phelps codes. Advances in Mathematics of Communications, 2010, 4 (1) : 69-81. doi: 10.3934/amc.2010.4.69 [11] Adrian Korban, Serap Şahinkaya, Deniz Ustun. A novel genetic search scheme based on nature-inspired evolutionary algorithms for binary self-dual codes. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022033 [12] Steven T. Dougherty, Esengül Saltürk, Steve Szabo. Codes over local rings of order 16 and binary codes. Advances in Mathematics of Communications, 2016, 10 (2) : 379-391. doi: 10.3934/amc.2016012 [13] Rafael Arce-Nazario, Francis N. Castro, Jose Ortiz-Ubarri. On the covering radius of some binary cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 329-338. doi: 10.3934/amc.2017025 [14] Stefka Bouyuklieva, Anton Malevich, Wolfgang Willems. On the performance of binary extremal self-dual codes. Advances in Mathematics of Communications, 2011, 5 (2) : 267-274. doi: 10.3934/amc.2011.5.267 [15] Daniel Heinlein, Sascha Kurz. Binary subspace codes in small ambient spaces. Advances in Mathematics of Communications, 2018, 12 (4) : 817-839. doi: 10.3934/amc.2018048 [16] Arezoo Soufi Karbaski, Taher Abualrub, Nuh Aydin, Peihan Liu. Additive polycyclic codes over $\mathbb{F}_{4}$ induced by binary vectors and some optimal codes. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022004 [17] Petr Lisoněk, Layla Trummer. Algorithms for the minimum weight of linear codes. Advances in Mathematics of Communications, 2016, 10 (1) : 195-207. doi: 10.3934/amc.2016.10.195 [18] Jean Creignou, Hervé Diet. Linear programming bounds for unitary codes. Advances in Mathematics of Communications, 2010, 4 (3) : 323-344. doi: 10.3934/amc.2010.4.323 [19] Steven Dougherty, Adrian Korban, Serap Șahinkaya, Deniz Ustun. Binary self-dual and LCD codes from generator matrices constructed from two group ring elements by a heuristic search scheme. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022036 [20] Fernando Hernando, Diego Ruano. New linear codes from matrix-product codes with polynomial units. Advances in Mathematics of Communications, 2010, 4 (3) : 363-367. doi: 10.3934/amc.2010.4.363

2021 Impact Factor: 1.015