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

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 ≤ d ≤ 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] [85,9, 40] 2 codes.

LinCode -- computer classification of linear codes

“…To turn these multitude of codes into something more manageable, we have used those results to classify all even [k + 10, k, 6] 2 codes. For k ≥ 12 their numbers are given by 127, 8, and 1, i.e., there is a unique even [24,14,6] 2 code, which is e.g. generated by For length n = 20 the most time expensive step, i.e., extending the [19,7,6] 2 codes to [20,8,6] 2 codes, took roughly 250 hours of computation time on a single core of a 2.80GHz laptop.…”

“…However, we are not 100% sure that in those mentioned cases, which run on the computing cluster, the stated numbers are correct, which makes it a perfect opportunity for independent verification by other algorithms. 8]; -the enumerations results for the uniqueness of the [46, 9, 20] 2 code presented in [14]; -the enumeration of the projective 2, 4-, and 8-divisible binary linear codes from [9]; -the counts of 9-divisible ternary codes in [5,Table 6]; and -the counts of 4-divisible quaternary codes in [5,Table 7]. Just to also have an extended example for a field size q > 2 we have extended the results from [5,Table 6] on 9-divisible ternary codes to dimensions k ≤ 8 and length n ≤ 55, see Table 4.…”

Divisible Codes

“…[5]. An interesting example is given by the [46,9,20] 2 -code found in [133]. It is optimal, unique, and does not have any non-trivial automorphism.…”

No Projective 16-Divisible Binary Linear Code of Length 131 Exists

“…In 2000 the determination of n(8, d) was completed in [2]. Not many of the open cases for n(9, d) have been resolved since then and we only refer to most recent paper [6].…”

“…Here a/b q r denotes the maximal integer t such that there exists a q r -divisible q-ary linear code of effective length n = a − tb and a code is called q r -divisible if the Hamming weights wt(c) of all codewords c are divisible by q r . For integers r the possible length of q r -divisible codes have been completely determined in [5] and except for the cases (n, d, k, q) = (6,4,3,2) and (8, 4, 3, 2) no tighter bound for A q (n, d; k) with d > 2k is known. For the case d = 2k, where the constantdimension codes are also called partial spreads, the notion of a/b q r can be sharpened by requiring the existence of a projective q r -divisible q-ary linear code of effective length n = a − tb.…”