The geometry of Hermitian self-orthogonal codes

TL;DR

This paper investigates the conditions under which linear codes over finite fields can be truncated or extended to become Hermitian self-orthogonal, revealing geometric and algebraic criteria based on code parameters and Hermitian forms.

Contribution

It establishes new bounds and criteria for when codes can be transformed into Hermitian self-orthogonal codes through truncation or extension, linking code geometry to Hermitian form conditions.

Findings

For n > k^2, codes have truncations equivalent to Hermitian self-orthogonal codes.

Truncations occur when generator matrix columns do not impose independent Hermitian form conditions.

Additional zeros of Hermitian forms lead to code extensions with Hermitian self-orthogonal truncations.

Abstract

We prove that if $n >k^2$ then a $k$-dimensional linear code of length $n$ over ${\mathbb F}_{q^2}$ has a truncation which is linearly equivalent to a Hermitian self-orthogonal linear code. In the contrary case we prove that truncations of linear codes to codes equivalent to Hermitian self-orthogonal linear codes occur when the columns of a generator matrix of the code do not impose independent conditions on the space of Hermitian forms. In the case that there are more than $n$ common zeros to the set of Hermitian forms which are zero on the columns of a generator matrix of the code, the additional zeros give the extension of the code to a code that has a truncation which is equivalent to a Hermitian self-orthogonal code.