How to Expand a Self-orthogonal Code

TL;DR

This work studies how to enlarge self-orthogonal codes while preserving their orthogonality. It develops constructive expansion procedures for both Hermitian and Euclidean self-orthogonal codes using Gram–Schmidt orthogonalization within appropriate dual spaces to increase the dimension by 1 at each step, forming code towers. For Hermitian self-orthogonal codes with $n>2k+1$, every such code lies in an $[n,k+1]_r$ Hermitian self-orthogonal code; for Euclidean self-orthogonal codes, expansions are possible under $n\ge 2k+3$ (or $n\ge 2k+2$ when $p=2$), with additional square-condition requirements in some boundary cases. The results yield termination either at Hermitian self-dual or almost self-dual codes (Hermitian case) or at self-dual codes under certain conditions (Euclidean case), and they discuss how the minimum distance behaves through expansions. The paper also sketches two feasible algorithms for performing these expansions and notes open questions on optimizing the resulting minimum distance.

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.

Theorems & Definitions (13)

• Theorem 1
• proof
• Corollary 1
• proof
• proposition 1
• proof
• Theorem 2
• proof
• Corollary 2
• proof