Divisibility of Griesmer Codes

TL;DR

This paper investigates the divisibility properties of Griesmer codes, establishing new conditions under which codeword weights are divisible by powers of the underlying prime power, and introduces a basis construction for these codes.

Contribution

It extends Ward's 1998 results by proving divisibility of codeword weights for q-powers and introduces a basis construction that reveals structural properties of Griesmer codes.

Findings

If q^e divides d, then p^e divides the weight of all codewords.

A basis exists such that certain subcodes are Griesmer codes with constant weight.

Divisibility of codeword weights by a factor Δ related to p^e and q.

Abstract

In this paper, we consider Griesmer codes, namely those linear codes meeting the Griesmer bound. Let $C$ be an $[n,k,d]_q$ Griesmer code with $q=p^f$, where $p$ is a prime and $f\ge1$ is an integer. In 1998, Ward proved that for $q=p$, if $p^e|d$, then $p^e|\mathrm{wt}(c)$ for all $c\in C$. In this paper, we show that if $q^e|d$, then $C$ has a basis consisting of $k$ codewords such that the first $\min\left\{e+1,k\right\}$ of them span a Griesmer subcode with constant weight $d$ and any $k-1$ of them span a $[g_q(k-1,d),k-1,d]_q$ Griesmer subcode. Using the $p$-adic algebraic method together with this basis, we prove that if $q^e|d$, then $p^e|\mathrm{wt}(c)$ for all $c\in C$. Based on this fact, using the geometric approach with the aforementioned basis, we show that if $p^e|d$, then $\Delta |{\rm wt}(c)$ for all $c\in C$, where $\Delta=\left\lceil p^{e-(f-1)(q-2)}\right\rceil$.