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

Sascha Kurz

doi:10.3934/amc.2020074

Article Contents

2021, Volume 15, Issue 3: 415-422. Doi: 10.3934/amc.2020074

This issue Previous Article Ironwood meta key agreement and authentication protocol Next Article

$ s $

-PD-sets for codes from projective planes

$ \mathrm{PG}(2,2^h) $

$ 5 \leq h\leq 9 $

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

Sascha Kurz

Mathematisches Institut, Universität Bayreuth, D-95440 Bayreuth, Germany

Received: September 2019

Revised: January 2020

Early access: April 2020

Published: August 2021

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Abstract

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.

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Mathematics Subject Classification: Primary: 94B65; Secondary: 94B05.

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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

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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

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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

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