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

## TL;DR

This paper constructs a unique optimal binary linear code of parameters [46,9,20], proves its uniqueness and asymmetry, and establishes the non-existence of certain related codes, advancing the understanding of code optimality and uniqueness.

## Contribution

It presents the first known [46,9,20]_2 code, proves its uniqueness and asymmetry, and shows non-existence of specific larger or related codes.

## Key findings

- Constructed a [46,9,20]_2 code and proved its uniqueness.
- Showed the code is asymmetric with a trivial automorphism group.
- Established non-existence of [47,10,20]_2 and [85,9,40]_2 codes.

## 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.