Cyclic codes and cyclically covering subspaces

Abstract

A subspace of ${\mathbb{F}}_q^n$ is called cyclically covering if the union of $\ddot{I}{ƒ}^i(U)$ can cover the whole space ${\mathbb{F}}_q^n$, where $\ddot{I}ƒ$ is the cyclic shift, $0⩽i⩽n-1$. Let $h_q(n)$ be the largest possible co-dimension of a cyclically covering subspace of ${\mathbb{F}}_q^n$. We show that $h_2(2p)=2$ for every prime $p$ such that $2$ is a primitive root modulo $p$. By constacyclic codes, we show that $h_q((q-1)n)=0$ when $h_q(n)=0$ and $\gcd (n,q-1)=1$. We also derive a lower bound on $h_q(n)$ by the concept of support weight distribution, which is important in coding theory. Finally, using irreducible cyclic codes, we present several families of $n$ such that $h_q(n)=0$.

20 pages