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

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 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∈(0,1), if a rate R code can be list-of-L decoded up to error fraction L/L+1 (1-R -ε), then its alphabet must have size at least exp(Ω_L,R(1/ε)). 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/ε^2) are unique-decodable up to error fraction (1-R-ε)/2. Our bounds hold even for subconstant ε≥ 1/n, implying that any code exactly achieving the L-th generalized Singleton bound requires alphabet size 2^Ω_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/ε))-sized alphabets, can be list-of-L decoded up to error fraction L/L+1 (1-R -ε).