American Institute of Mathematical Sciences

Advanced
doi: 10.3934/amc.2020074

The $[46, 9, 20]_2$ code is unique

Sascha Kurz

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.

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, 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

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.  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

Table 1.  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
Table Options

Download as excel

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
Table Options

Download as excel

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
Table 3.  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
Table Options

Download as excel

$ A_{56} $ 3 4 5 6 7 8 9
25773 48792 26091 5198 450 17 1
[1]

Vito Napolitano, Ferdinando Zullo. Codes with few weights arising from linear sets. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020129
[2]

Dandan Wang, Xiwang Cao, Gaojun Luo. A class of linear codes and their complete weight enumerators. Advances in Mathematics of Communications, 2021, 15 (1) : 73-97. doi: 10.3934/amc.2020044
[3]

Nicola Pace, Angelo Sonnino. On the existence of PD-sets: Algorithms arising from automorphism groups of codes. Advances in Mathematics of Communications, 2021, 15 (2) : 267-277. doi: 10.3934/amc.2020065
[4]

Shudi Yang, Xiangli Kong, Xueying Shi. Complete weight enumerators of a class of linear codes over finite fields. Advances in Mathematics of Communications, 2021, 15 (1) : 99-112. doi: 10.3934/amc.2020045
[5]

Jong Yoon Hyun, Boran Kim, Minwon Na. Construction of minimal linear codes from multi-variable functions. Advances in Mathematics of Communications, 2021, 15 (2) : 227-240. doi: 10.3934/amc.2020055
[6]

Shanding Xu, Longjiang Qu, Xiwang Cao. Three classes of partitioned difference families and their optimal constant composition codes. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020120
[7]

Xiangrui Meng, Jian Gao. Complete weight enumerator of torsion codes. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020124
[8]

Fengwei Li, Qin Yue, Xiaoming Sun. The values of two classes of Gaussian periods in index 2 case and weight distributions of linear codes. Advances in Mathematics of Communications, 2021, 15 (1) : 131-153. doi: 10.3934/amc.2020049
[9]

Hongming Ru, Chunming Tang, Yanfeng Qi, Yuxiao Deng. A construction of $ p $-ary linear codes with two or three weights. Advances in Mathematics of Communications, 2021, 15 (1) : 9-22. doi: 10.3934/amc.2020039
[10]

Agnaldo José Ferrari, Tatiana Miguel Rodrigues de Souza. Rotated $ A_n $-lattice codes of full diversity. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020118
[11]

San Ling, Buket Özkaya. New bounds on the minimum distance of cyclic codes. Advances in Mathematics of Communications, 2021, 15 (1) : 1-8. doi: 10.3934/amc.2020038
[12]

Karan Khathuria, Joachim Rosenthal, Violetta Weger. Encryption scheme based on expanded Reed-Solomon codes. Advances in Mathematics of Communications, 2021, 15 (2) : 207-218. doi: 10.3934/amc.2020053
[13]

Sabira El Khalfaoui, Gábor P. Nagy. On the dimension of the subfield subcodes of 1-point Hermitian codes. Advances in Mathematics of Communications, 2021, 15 (2) : 219-226. doi: 10.3934/amc.2020054
[14]

Tingting Wu, Li Liu, Lanqiang Li, Shixin Zhu. Repeated-root constacyclic codes of length $ 6lp^s $. Advances in Mathematics of Communications, 2021, 15 (1) : 167-189. doi: 10.3934/amc.2020051
[15]

Saadoun Mahmoudi, Karim Samei. Codes over $ \frak m $-adic completion rings. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020122
[16]

Hongwei Liu, Jingge Liu. On $ \sigma $-self-orthogonal constacyclic codes over $ \mathbb F_{p^m}+u\mathbb F_{p^m} $. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020127
[17]

Ivan Bailera, Joaquim Borges, Josep Rifà. On Hadamard full propelinear codes with associated group $ C_{2t}\times C_2 $. Advances in Mathematics of Communications, 2021, 15 (1) : 35-54. doi: 10.3934/amc.2020041
[18]

Yuan Cao, Yonglin Cao, Hai Q. Dinh, Ramakrishna Bandi, Fang-Wei Fu. An explicit representation and enumeration for negacyclic codes of length $ 2^kn $ over $ \mathbb{Z}_4+u\mathbb{Z}_4 $. Advances in Mathematics of Communications, 2021, 15 (2) : 291-309. doi: 10.3934/amc.2020067
[19]

Hai Q. Dinh, Bac T. Nguyen, Paravee Maneejuk. Constacyclic codes of length $ 8p^s $ over $ \mathbb F_{p^m} + u\mathbb F_{p^m} $. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020123
[20]

Denis Serre. Non-linear electromagnetism and special relativity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 435-454. doi: 10.3934/dcds.2009.23.435

2019 Impact Factor: 0.734

Tools

Metrics

  • PDF downloads (42)
  • HTML views (359)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences