A Griesmer-Type Bound for List-Decodable Linear Codes

2026-07-21

AI Summary

This study investigates the fundamental limits of list-decodable linear codes under a fixed list size, aiming to surpass classical Singleton-type bounds. By introducing generalized Hamming weights into the analysis of list decoding for the first time and leveraging ideas from the Griesmer bound, the authors derive a lower bound on block length for small list sizes and establish a tighter Griesmer-type bound. A key contribution is the proof that for $1 \leq L \leq q-1$, any $(\tau,L)$-list-decodable $q$-ary linear code must have minimum distance at least $\tau + \lfloor \tau/L \rfloor + 1$. Furthermore, for $q = 3^a$, they explicitly construct a $[q+3,2,q+1]$ code meeting this new bound and achieving $(2q/3,2)$-list decodability.

Abstract

A code $C\subseteq F_q^n$ is $(τ,L)$-list-decodable if every Hamming ball of radius $τ$ contains at most $L$ codewords of $C$. Here $τ$ is the list-decoding radius, and $L$ is the list size. Singleton-type bounds constrain the radius and the rate when $L$ is fixed. These bounds are not the only possible constraints on list-decodable codes. In this paper, we derive an upper bound on the list-decoding radius in terms of generalized Hamming weights. As a consequence, for $1\le L\le q-1$, every $(τ,L)$-list-decodable $q$-ary linear code has minimum distance at least $ τ+\left\lfloor \fracτ{L}\right\rfloor+1. $ Combining this lower bound with the classical Griesmer bound gives a Griesmer-type lower bound on the block length. For $q=3^a$, we construct an explicit family of $q$-ary linear $[q+3,2,q+1]$ codes. These codes are $(2q/3,2)$-list-decodable and meet the Griesmer-type bound with equality. They do not attain the Singleton-type bound. Thus the Griesmer-type bound can be a strict improvement over the Singleton-type bound.

Problem

• list-decodable codes
• Griesmer bound
• linear codes
• generalized Hamming weights
• Singleton-type bound

Innovation

• list-decodable codes
• Griesmer bound
• generalized Hamming weights
• linear codes
• Singleton-type bound