Distance Distributions of Cyclic Orbit Codes

TL;DR

This paper analyzes the structure of distance distributions in cyclic orbit codes, revealing dependencies on parameters and providing partial characterizations for certain cases, advancing understanding of their combinatorial properties.

Contribution

It characterizes the distance distribution of optimal full-length cyclic orbit codes and offers partial results for codes with lower minimum distance.

Findings

Distance distribution depends only on q, n, and the subspace dimension for optimal full-length codes.

Partial characterization of distance distribution for codes with small intersection dimension.

Brief analysis of distance distribution for unions of optimal orbit codes.

Abstract

The distance distribution of a code is the vector whose $i^\text{th}$ entry is the number of pairs of codewords with distance $i$. We investigate the structure of the distance distribution for cyclic orbit codes, which are subspace codes generated by the action of $\mathbb{F}_{q^n}^*$ on an $\mathbb{F}_q$-subspace $U$ of $\mathbb{F}_{q^n}$. We show that for optimal full-length orbit codes the distance distribution depends only on $q,\,n$, and the dimension of $U$. For full-length orbit codes with lower minimum distance, we provide partial results towards a characterization of the distance distribution, especially in the case that any two codewords intersect in a space of dimension at most 2. Finally, we briefly address the distance distribution of a union of optimal full-length orbit codes.