On the lower bound for the length of minimal codes

TL;DR

This paper investigates lower bounds on the length of minimal codes over finite fields, establishing new asymptotic bounds and structural properties, especially for binary codes, revealing that previous bounds are not tight for large dimensions.

Contribution

It proves a new asymptotic lower bound on minimal code length growth and provides structural insights for binary minimal codes, improving understanding of their minimal lengths.

Findings

Asymptotic lower bound: m(k,q)/k ≥ (q + ε(q)) with ε(q) > 1.52 for q=2

Structural results for minimal binary codes of length 3(k-1)

Identification of cases where previous bounds are not tight

Abstract

In recent years, many connections have been made between minimal codes, a classical object in coding theory, and other remarkable structures in finite geometry and combinatorics. One of the main problems related to minimal codes is to give lower and upper bounds on the length $m(k,q)$ of the shortest minimal codes of a given dimension $k$ over the finite field $\mathbb{F}_q$. It has been recently proved that $m(k, q) \geq (q+1)(k-1)$. In this note, we prove that $\liminf_{k \rightarrow \infty} \frac{m(k, q)}{k} \geq (q+ \varepsilon(q) )$, where $\varepsilon$ is an increasing function such that $1.52 <\varepsilon(2)\leq \varepsilon(q) \leq \sqrt{2} + \frac{1}{2}$. Hence, the previously known lower bound is not tight for large enough $k$. We then focus on the binary case and prove some structural results on minimal codes of length $3(k-1)$. As a byproduct, we are able to show that, if $k = 5 \pmod 8$ and for other small values of $k$, the bound is not tight.