A Griesmer-Type Bound for List-Decodable Linear Codes

Abstract

A code $C\subseteq F_q^n$ is $(\tau,L)$-list-decodable if every Hamming ball of radius $\tau$ contains at most $L$ codewords of $C$. Here $\tau$ 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 $(\tau,L)$-list-decodable $q$-ary linear code has minimum distance at least $ \tau+\left\lfloor \frac{\tau}{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.