Spread Codes from Abelian non-cyclic groups

TL;DR

The paper addresses constructing $k$-spreads in $F_q^n$ by exploiting orbits under a non-cyclic Abelian subgroup of $GL_n(F_q)$. It develops a two-stage approach: first, build a maximum-distance orbit code of dimension $k$ using a carefully designed Abelian non-cyclic group $H$ inside $GL_s(F_{q^k})$ and field reduction to obtain $k$-partial spreads; second, complete each partial spread by adjoining two orbital families to obtain a Desarguesian $k$-spread, yielding a full spread with a noncyclic orbital structure. The construction works for even $n=ks$ with $s=2t$ and gcd$(t,q^k-1)=1$ and extends prior cyclic and non-cyclic Abelian orbit code results, providing a new algebraic approach to spread construction in the Grassmannian. The results provide a new algebraic mechanism for producing large, well-structured constant-dimension codes with direct relevance to network coding and Grassmannian geometry, via explicit group actions and field-reduction techniques.

Abstract

Given the finite field $\mathbb{F}_{q}$, for a prime power $q$, in this paper we present a way of constructing spreads of $\mathbb{F}_{q}^{n}$. They will arise as orbits under the action of an Abelian non-cyclic group. First, we construct a family of orbit codes of maximum distance using this group, and then we complete each of these codes to achieve a spread of the whole space having an orbital structure.

Theorems & Definitions (19)

• Lemma 1
• proof
• Lemma 2
• Lemma 3
• proof
• Theorem 1
• proof
• Remark 1
• Lemma 4
• proof