Subspace codes in PG(2n-1,q)

TL;DR

This paper constructs large constant-dimension subspace codes in projective geometry, significantly improving known sizes for certain parameters, with applications in network coding error correction.

Contribution

It introduces new constructions of subspace codes in PG(2n-1,q) that surpass existing sizes, especially for n > 4, and provides explicit code sizes for specific cases.

Findings

Constructed $(2n,M,4;n)_q$ codes for all $n \,\geq\, 4$ with larger sizes.

Achieved a new code with size $M = q^{12}+q^2(q^2+1)^2(q^2+q+1)+1$ for $n=4$.

Codes have potential applications in error correction for network coding.

Abstract

An $(r,M,2\delta;k)_q$ constant--dimension subspace code, $\delta >1$, is a collection $\cal C$ of $(k-1)$--dimensional projective subspaces of ${\rm PG(r-1,q)}$ such that every $(k-\delta)$--dimensional projective subspace of ${\rm PG(r-1,q)}$ is contained in at most a member of $\cal C$. Constant--dimension subspace codes gained recently lot of interest due to the work by Koetter and Kschischang, where they presented an application of such codes for error-correction in random network coding. Here a $(2n,M,4;n)_q$ constant--dimension subspace code is constructed, for every $n \ge 4$. The size of our codes is considerably larger than all known constructions so far, whenever $n > 4$. When $n=4$ a further improvement is provided by constructing an $(8,M,4;4)_q$ constant--dimension subspace code, with $M = q^{12}+q^2(q^2+1)^2(q^2+q+1)+1$.