Improved AntiGriesmer Bounds for Linear Anticodes and Applications

TL;DR

This paper extends the antiGriesmer bound for linear anticodes by removing length restrictions and relaxing dual distance conditions, thereby broadening its applicability and providing new bounds and insights for code construction.

Contribution

It generalizes the antiGriesmer bound for linear anticodes, removing previous restrictions and establishing a more versatile inequality applicable to a wider class of codes.

Findings

The new bound applies without length restrictions.

It provides tighter bounds on code parameters.

Examples show the bound can be sharper than previous results.

Abstract

This paper improves the antiGriesmer bound for linear anticodes previously established by Chen and Xie (Journal of Algebra, 673 (2025) 304-320). While the original bound required the code length to satisfy $n < q^{k-1}$ and the dual code to have minimum distance at least 3, our main result removes the length restriction and relaxes the dual distance condition to at least 2. Specifically, we prove that for any $[n,k]_q$ linear anticode $\mathcal{C}$ over $\mathbb{F}_q$ with diameter $\delta$ and $d(\mathcal{C}^\perp) \geq 2$, the inequality \[ n \leq \sum_{i=0}^{k-1} \left\lfloor \frac{\delta}{q^i} \right\rfloor \] holds. This generalization significantly broadens the applicability of the antiGriesmer bound. We derive several corollaries, including lower bounds on the diameter $\delta$ in terms of $n$ and $k$, upper bounds on the code length $n$, and constraints on the dimension $k$. Applications to the construction and classification of linear codes with few weights are also discussed, along with examples demonstrating that our new bound can be sharper than previous ones. Our work unifies and extends earlier findings, providing a more comprehensive framework for studying linear anticodes and their properties.