Divisibility of Trace Codes

2026-05-19

AI Summary

This study investigates the $p$-adic valuation of trace codes, defined as the largest exponent such that the Hamming weight of every codeword is divisible by a power of a prime $p$. By extending Ward’s classical divisibility criterion from standard generator matrices to generalized generator matrices over extension fields, the authors establish a systematic framework applicable to trace codes. This approach constitutes the first extension of Ward’s criterion within the generalized generator matrix setting and is applied to analyze the $p$-adic valuations of Abelian codes and the number of solutions to Artin–Schreier-type equations. Key contributions include a concise proof of the divisibility property for Abelian codes, an exact determination of the minimal $p$-adic valuation of solution counts for Artin–Schreier equations under homogeneous polynomial conditions, and several explicit lower bounds.

Abstract

A linear code is said to be $Δ$-divisible if the Hamming weights of all its codewords are divisible by $Δ$. 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 $d$.

Problem

• divisible codes
• trace codes
• p-adic valuation
• generalized generator matrices
• Artin-Schreier equations

Innovation

• trace codes
• p-adic valuation
• divisibility criterion
• generalized generator matrices
• Artin-Schreier equations