Distance Distributions of Cyclic Orbit Codes

Abstract

The distance distribution of a code is the vector whose ith 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 ${\mathbb {F}}_{q^n}$. Note that ${\mathbb {F}}_{q^n}^*$ is a Singer cycle in the general linear group of all ${\mathbb {F}}_q$-automorphisms of ${\mathbb {F}}_{q^n}$. We show that for full-length orbit codes with maximal possible distance 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 full-length orbit codes with maximum distance.