Divisibility of Trace Codes
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 pt-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 semisimple 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 𝔽qm (where q = pe) to the Artin--Schreier type equation $ f(x_1,\ldots,x_k)=y^q-y $. In particular, under the coprime condition (d,(qm−1)/(q−1))=1, we determine the exact minimum p-adic valuation of the number of solutions when f is restricted to homogeneous polynomials of degree d.
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BibTeX
@misc{huang2026divisibility,
title = {{Divisibility of Trace Codes}},
author = {Hexiang Huang and Haihua Deng and Sihuang Hu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.19857},
eprint = {2605.19857},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.19857},
}
RIS
TY - UNPB
TI - Divisibility of Trace Codes
AU - Hexiang Huang
AU - Haihua Deng
AU - Sihuang Hu
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2605.19857
UR - https://arxiv.org/abs/2605.19857
AB - 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 semisimple 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 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 coprime condition (d,(q^m−1)/(q−1))=1, we determine the exact minimum p-adic valuation of the number of solutions when f is restricted to homogeneous polynomials of degree d.
ER -