Characterization and construction of optimal binary linear codes with one-dimensional hull

TL;DR

This paper investigates binary linear codes with one-dimensional hulls, establishing their properties and relation to LCD codes, and determines bounds on their maximum minimum distances for various parameters.

Contribution

It characterizes binary linear codes with one-dimensional hulls, relates them to LCD codes, and determines bounds on their maximum minimum distances for specific code lengths and dimensions.

Findings

Established properties of binary linear codes with one-dimensional hulls.

Derived inequalities relating hull dimensions to code parameters.

Determined maximum minimum distances for certain code lengths and dimensions.

Abstract

The hull of a linear code over finite fields is the intersection of the code and its dual, and linear codes with small hulls have applications in computational complexity and information protection. Linear codes with the smallest hull are LCD codes, which have been widely studied. Recently, several papers were devoted to related LCD codes over finite fields with size greater than 3 to linear codes with one-dimensional or higher dimensional hull. Therefore, an interesting and non-trivial problem is to study binary linear codes with one-dimensional hull with connection to binary LCD codes. The objective of this paper is to study some properties of binary linear codes with one-dimensional hull, and establish their relation with binary LCD codes. Some interesting inequalities are thus obtained. Using such a characterization, we study the largest minimum distance $d_{one}(n,k)$ among all binary linear $[n,k]$ codes with one-dimensional hull. We determine the largest minimum distances $d_{one}(n,n-k)$ for $ k\leq 5$ and $d_{one}(n,k)$ for $k\leq 4$ or $14\leq n\leq 24$. We partially determine the exact value of $d_{one}(n,k)$ for $k=5$ or $25\leq n\leq 30$.