A tight upper bound on the number of non-zero weights of a quasi-cyclic code

TL;DR

This paper derives explicit formulas and upper bounds for the number of non-zero weights in quasi-cyclic codes, improving existing bounds and providing conditions for codes that meet these bounds.

Contribution

It introduces a new explicit formula for automorphism group orbits and tighter bounds for non-zero weights in quasi-cyclic codes, generalizing previous results.

Findings

Explicit orbit formulas for automorphism groups

Tighter upper bounds on non-zero weights

Examples demonstrating bound tightness

Abstract

Let $\mathcal{C}$ be a quasi-cyclic code of index $l(l\geq2)$. Let $G$ be the subgroup of the automorphism group of $\mathcal{C}$ generated by $\rho^l$ and the scalar multiplications of $\mathcal{C}$, where $\rho$ denotes the standard cyclic shift. In this paper, we find an explicit formula of orbits of $G$ on $\mathcal{C}\setminus \{\mathbf{0}\}$. Consequently, an explicit upper bound on the number of nonzero weights of $\mathcal{C}$ is immediately derived and a necessary and sufficient condition for codes meeting the bound is exhibited. If $\mathcal{C}$ is a one-generator quasi-cyclic code, a tighter upper bound on the number of nonzero weights of $\mathcal{C}$ is obtained by considering a larger automorphism subgroup which is generated by the multiplier, $\rho^l$ and the scalar multiplications of $\mathcal{C}$. In particular, we list some examples to show the bounds are tight. Our main result improves and generalizes some of the results in \cite{M2}.