Cyclic codes and cyclically covering subspaces

2026-07-27

AI Summary

This study addresses the problem of determining the maximum codimension of cyclic covering subspaces over finite fields, which concerns the structural limits of subspaces whose cyclic shifts span the entire space. By leveraging algebraic coding theory—particularly cyclic and constacyclic codes together with support weight distributions—the work establishes, for the first time, exact values and lower bounds for this codimension and uncovers its deep connection to number-theoretic conditions, such as 2 being a primitive root modulo a prime $p$. The main contributions include proving that $h_2(2p) = 2$ when $p$ is an odd prime and 2 is a primitive root modulo $p$, providing sufficient conditions for $h_q((q-1)n) = 0$, and explicitly constructing several infinite families of integers $n$ for which $h_q(n) = 0$.

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 $h_q(n) = 0$.

Problem

• cyclic codes
• cyclically covering subspaces
• co-dimension
• finite fields
• support weight distribution

Innovation

• cyclically covering subspace
• constacyclic codes
• support weight distribution
• irreducible cyclic codes
• co-dimension