English

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

Combinatorics 2020-04-15 v2 Information Theory math.IT

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=46n=46 with known bounds 19d2019\le d\le 20. Here we present a [46,9,20]2[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[47,10,20]_2 and [85,9,40]2[85,9,40]_2 codes.

Keywords

Cite

@article{arxiv.1906.02621,
  title  = {The ${[46,9,20]_2}$ code is unique},
  author = {Sascha Kurz},
  journal= {arXiv preprint arXiv:1906.02621},
  year   = {2020}
}

Comments

7 pages, 3 tables, typos corrected

R2 v1 2026-06-23T09:45:29.357Z