How to Expand a Self-orthogonal Code

TL;DR

This paper presents methods to expand Euclidean and Hermitian self-orthogonal codes while preserving their orthogonal properties, including algorithms for practical implementation and conditions for successful expansion.

Contribution

It introduces new procedures for expanding self-orthogonal codes and provides algorithms and conditions for their successful application.

Findings

Every $k$-dimensional Hermitian self-orthogonal code is contained in a $(k+1)$-dimensional code.

For $k< n/2-1$, every $[n,k]$ Euclidean self-orthogonal code can be expanded to $[n,k+1]$.

Expansion is possible under specific conditions for $k=n/2-1$ depending on the prime $p$.

Abstract

In this paper, we show how to expand Euclidean/Hermitian self-orthogonal code preserving their orthogonal property. Our results show that every $k$-dimension Hermitian self-orthogonal code is contained in a $(k+1)$-dimensional Hermitian self-orthogonal code. Also, for $k< n/2-1$, every $[n,k]$ Euclidean self-orthogonal code is contained in an $[n,k+1]$ Euclidean self-orthogonal code. Moreover, for $k=n/2-1$ and $p=2$, we can also fulfill the expanding process. But for $k=n/2-1$ and $p$ odd prime, the expanding process can be fulfilled if and only if an extra condition must be satisfied. We also propose two feasible algorithms on these expanding procedures.