Optimal Binary $(5,3)$ Projective Space Codes from Maximal Partial Spreads

TL;DR

This paper explores new constructions of optimal binary (5,3) projective space codes by combining minimal change strategies with spread structure modifications, resulting in codes outside existing frameworks.

Contribution

It introduces novel binary (5,3) codes using a combined approach of minimal changes and spread modifications, expanding beyond current construction methods.

Findings

Several new optimal binary (5,3) codes were constructed.

The new codes differ structurally from existing constructions.

Examples demonstrate codes outside the known framework.

Abstract

Recently a construction of optimal non-constant dimension subspace codes, also termed projective space codes, has been reported in a paper of Honold-Kiermaier-Kurz. Restricted to binary codes in a 5-dimensional ambient space with minimum subspace distance 3, these optimal codes were interpreted in terms of maximal partial spreads of 2-dimensional subspaces. In a parallel development, an optimal binary (5,3) code was obtained by a minimal change strategy on a nearly optimal example of Etzion and Vardy. In this article, we report several examples of optimal binary (5,3) codes obtained by the application of this strategy combined with changes to the spread structure of existing codes. We also establish that, based on the types of constituent spreads, our examples lie outside the framework of the existing construction.