Constructions and Bounds for Mixed-Dimension Subspace Codes

TL;DR

This paper addresses the maximum size of subspace codes in finite projective spaces for certain minimum distances, providing exact values, improved bounds, and comprehensive tables for small parameters.

Contribution

It completely solves the maximum size problem for subspace codes when the minimum distance is at least v-1 and offers new bounds and exact values for specific cases.

Findings

Resolved the maximum size for d ≥ v-1.

Determined A_q(5,3)=2q^3+2 and A_2(7,5)=34.

Provided bounds and tables for A_2(v,d) for v ≤ 7.

Abstract

Codes in finite projective spaces equipped with the subspace distance have been proposed for error control in random linear network coding. The resulting so-called \emph{Main Problem of Subspace Coding} is to determine the maximum size $A_q(v,d)$ of a code in $\operatorname{PG}(v-1,\mathbb{F}_q)$ with minimum subspace distance $d$. Here we completely resolve this problem for $d\ge v-1$. For $d=v-2$ we present some improved bounds and determine $A_q(5,3)=2q^3+2$ (all $q$), $A_2(7,5)=34$. We also provide an exposition of the known determination of $A_q(v,2)$, and a table with exact results and bounds for the numbers $A_2(v,d)$, $v\leq 7$.