Divisibility of Trace Codes

TL;DR

This paper introduces a divisibility criterion for trace codes, extending classical results and providing bounds on solutions to Artin-Schreier equations over finite fields.

Contribution

It develops a systematic method to determine the p-adic valuation of trace codes, generalizing Ward's divisibility criterion to broader generator matrices.

Findings

Provides a concise proof of divisibility results for abelian codes.

Establishes explicit lower bounds on p-adic valuations of solutions to Artin-Schreier equations.

Determines the exact minimum p-adic valuation under certain conditions.

Abstract

A linear code is said to be $\Delta$-divisible if the Hamming weights of all its codewords are divisible by $\Delta$. The $p$-adic valuation of a code is defined as the greatest integer $t$ such that the code is $p^t$-divisible. In this paper, we establish a divisibility criterion for trace codes. Specifically, this criterion provides a systematic method to determine the $p$-adic valuation of the associated trace code, thereby extending Ward's classical divisibility criterion from standard generating sets (or matrices) to generalized generator matrices over an extension field. Furthermore, we present two applications of our framework. The first application provides a concise proof of the celebrated divisibility results on abelian codes established by Delsarte and McEliece. The second application establishes several explicit lower bounds on the $p$-adic valuation of the number of solutions over $\mathbb{F}_{q^m}$ (where $q = p^e$) to the Artin-Schreier type equation $ f(x_1,\ldots,x_k)=y^q-y $. In particular, under the condition $\left(d,\frac{q^m-1}{q-1}\right)=1$, we determine the exact minimum $p$-adic valuation of the number of solutions when $f$ is restricted to homogeneous polynomials of degree $d$.