Improved AntiGriesmer Bounds for Linear Anticodes and Applications
Guanghui Zhang, Bocong Chen, Liren Lin, Hongwei Liu
TL;DR
The paper broadens the antiGriesmer bound for linear anticodes by removing the length restriction $n<q^{k-1}$ and weakening $d(\mathcal C^{\perp})\ge 3$ to $d(\mathcal C^{\perp})\ge 2$. It proves the central bound $n \le \sum_{i=0}^{k-1} \left\lfloor \dfrac{δ}{q^i} \right\rfloor$ for any $[n,k]_q$ linear anticode with diameter $δ$, using a constructive partitioning argument via nonzero linear functionals and a projective counting technique. The work yields corollaries that bound $δ$ in terms of $n$ and $k$, bound $n$ in terms of $δ$ and $k$, and relate $n$, $w$, and $δ$ through weight-based refinements, including comparisons to prior CX results and examples showing sharpness. Overall, the results unify and extend earlier findings, broadening the applicability of antiGriesmer-type bounds and enhancing the construction and classification of linear anticodes and related few-weight codes over finite fields.
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 $δ$ and $d(\mathcal{C}^\perp) \geq 2$, the inequality \[ n \leq \sum_{i=0}^{k-1} \left\lfloor \fracδ{q^i} \right\rfloor \] holds. This generalization significantly broadens the applicability of the antiGriesmer bound. We derive several corollaries, including lower bounds on the diameter $δ$ 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.