AG codes have no list-decoding friends: Approaching the generalized Singleton bound requires exponential alphabets

TL;DR

This paper proves that approaching the generalized Singleton bound for list-decoding requires exponentially large alphabets, contrasting with unique decoding where smaller alphabets suffice, highlighting fundamental limits in coding theory.

Contribution

The authors establish exponential lower bounds on alphabet size for list-decoding near the generalized Singleton bound, extending previous results and showing tightness with random codes.

Findings

Exponential alphabet size is necessary for approaching the list-decoding bound.

Existing algebraic-geometry codes over small alphabets cannot approach the bound.

Random codes over exponential-sized alphabets can achieve near-bound list-decoding.

Abstract

A simple, recently observed generalization of the classical Singleton bound to list-decoding asserts that rate $R$ codes are not list-decodable using list-size $L$ beyond an error fraction $\frac{L}{L+1} (1-R)$ (the Singleton bound being the case of $L=1$, i.e., unique decoding). We prove that in order to approach this bound for any fixed $L >1$, one needs exponential alphabets. Specifically, for every $L>1$ and $R\in(0,1)$, if a rate $R$ code can be list-of-$L$ decoded up to error fraction $\frac{L}{L+1} (1-R -\varepsilon)$, then its alphabet must have size at least $\exp(\Omega_{L,R}(1/\varepsilon))$. This is in sharp contrast to the situation for unique decoding where certain families of rate $R$ algebraic-geometry (AG) codes over an alphabet of size $O(1/\varepsilon^2)$ are unique-decodable up to error fraction $(1-R-\varepsilon)/2$. Our bounds hold even for subconstant $\varepsilon\ge 1/n$, implying that any code exactly achieving the $L$-th generalized Singleton bound requires alphabet size $2^{\Omega_{L,R}(n)}$. Previously this was only known only for $L=2$ under the additional assumptions that the code is both linear and MDS. Our lower bound is tight up to constant factors in the exponent -- with high probability random codes (or, as shown recently, even random linear codes) over $\exp(O_L(1/\varepsilon))$-sized alphabets, can be list-of-$L$ decoded up to error fraction $\frac{L}{L+1} (1-R -\varepsilon)$.