New upper bounds on binary linear codes and a $\mathbb Z_4$-code with a better-than-linear Gray image

TL;DR

This paper establishes new upper bounds for binary linear codes using integer linear programming, and demonstrates a $ ext{Z}_4$-linear code with a Gray image surpassing all binary linear codes of similar parameters.

Contribution

It introduces novel non-existence results for certain binary linear codes and highlights a $ ext{Z}_4$-linear code with a superior Gray image, expanding the understanding of code optimality.

Findings

No binary linear [1988, 12, 992] code exists.

Certain other binary linear codes with specified parameters do not exist.

A $ ext{Z}_4$-linear code with a better-than-linear Gray image is identified.

Abstract

Using integer linear programming and table-lookups we prove that there is no binary linear $[1988, 12, 992]$ code. As a by-product, the non-existence of binary linear codes with the parameters $[324, 10, 160]$, $[356, 10, 176]$, $[772,11,384]$, and $[836,11,416]$ is shown. Our work is motivated by the recent construction of the extended dualized Kerdock code $\hat{\mathcal{K}}^*_{6}$, which is a $\mathbb{Z}_4$-linear code having a non-linear binary Gray image with the parameters $(1988,2^{12},992)$. By our result, the code $\hat{\mathcal{K}}^*_{6}$ can be added to the small list of $\mathbb{Z}_4$-codes for which it is known that the Gray image is better than any binary linear code.