Equidistant Linear Codes in Projective Spaces

TL;DR

This paper investigates equidistant linear codes in projective spaces, establishing bounds on their size, structural properties, and existence for all prime powers, thereby advancing the understanding of their combinatorial and geometric characteristics.

Contribution

It proves the maximum size of such codes for q=2, characterizes their structure, and demonstrates their existence for all prime powers using Steiner triple systems.

Findings

Maximum size of equidistant linear codes for q=2 is 2^n.

Codes attaining maximum size relate to Fano plane and sunflower structures.

Existence of equidistant linear codes for any prime power q using Steiner triple systems.

Abstract

Linear codes in the projective space $\mathbb{P}_q(n)$, the set of all subspaces of the vector space $\mathbb{F}_q^n$, were first considered by Braun, Etzion and Vardy. The Grassmannian $\mathbb{G}_q(n,k)$ is the collection of all subspaces of dimension $k$ in $\mathbb{P}_q(n)$. We study equidistant linear codes in $\mathbb{P}_q(n)$ in this paper and establish that the normalized minimum distance of a linear code is maximum if and only if it is equidistant. We prove that the upper bound on the size of such class of linear codes is $2^n$ when $q=2$ as conjectured by Braun et al. Moreover, the codes attaining this bound are shown to have structures akin to combinatorial objects, viz. \emph{Fano plane} and \emph{sunflower}. We also prove the existence of equidistant linear codes in $\mathbb{P}_q(n)$ for any prime power $q$ using \emph{Steiner triple system}. Thus we establish that the problem of finding equidistant linear codes of maximum size in $\mathbb{P}_q(n)$ with constant distance $2d$ is equivalent to the problem of finding the largest $d$-intersecting family of subspaces in $\mathbb{G}_q(n, 2d)$ for all $1 \le d \le \lfloor \frac{n}{2}\rfloor$. Our discovery proves that there exist equidistant linear codes of size more than $2^n$ for every prime power $q > 2$.