Cyclic codes and cyclically covering subspaces
2026-07-27 • Information Theory
Information Theory
AI summaryⓘ
The authors study special subspaces in vector spaces over finite fields that can cover the whole space by repeatedly shifting their elements cyclically. They define a measure, h_q(n), to capture how large the co-dimension of such a subspace can be. They prove exact values for h_2(2p) when 2 is a primitive root modulo a prime p, explore conditions when h_q(n) equals zero using code theory concepts, and provide lower bounds based on support weight distributions. The paper also identifies families of dimensions n for which this measure is zero by using irreducible cyclic codes.
finite fieldvector spacecyclic shiftcyclically covering subspaceco-dimensionprimitive rootconstacyclic codesupport weight distributionirreducible cyclic codecoding theory
Authors
Xuan Wang, Minjia Shi
Abstract
A subspace of $\mathbb{F}_q^n$ is called cyclically covering if the union of $σ^i(U)$ can cover the whole space $\mathbb{F}_q^n$, where $σ$ is the cyclic shift, $0 \leqslant i \leqslant n-1$. 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 $h_q(n) = 0$.