A Griesmer-Type Bound for List-Decodable Linear Codes
Provides a new fundamental bound for list-decodable codes, offering a tighter constraint than the Singleton-type bound for certain parameters, which is of interest to coding theorists.
The paper derives a Griesmer-type bound for list-decodable linear codes, showing that for 1 ≤ L ≤ q-1, the minimum distance must be at least τ + ⌊τ/L⌋ + 1. They construct explicit q-ary linear [q+3,2,q+1] codes that are (2q/3,2)-list-decodable and meet this bound with equality, demonstrating it can be stricter than the Singleton-type bound.
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.