Enumeration of linear codes with different hulls

Author:

Stefka Bouyuklieva, Iliya Bouyukliev

Keyword:

Computer Science, Information Theory, Information Theory (cs.IT), Discrete Mathematics (cs.DM)

journal:

date:

2024-02-02 00:00:00

Abstract

The hull of a linear code $C$ is the intersection of $C$ with its dual code. We present and analyze the number of linear $q$-ary codes of the same length and dimension but with different dimensions for their hulls. We prove that for given dimension $k$ and length $n\ge 2k$ the number of all $[n,k]_q$ linear codes with hull dimension $l$ decreases as $l$ increases. We also present classification results for binary and ternary linear codes with trivial hulls (LCD and self-orthogonal) for some values of the length $n$ and dimension $k$, comparing the obtained numbers with the number of all linear codes for the given $n$ and $k$.

PDF: Enumeration of linear codes with different hulls.pdf