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 $\tfrac{L}{L+1} (1-R)$ (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 $\tfrac{L}{L+1} (1-R -\varepsilon)$, 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 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 $\tfrac{L}{L+1} (1-R -\varepsilon)$.

翻译：关于列表解码的经典Singleton界的一个简单且近期被推广的结论表明：对于速率$R$的码，在列表大小$L$的限制下，无法对超出误差分数$\tfrac{L}{L+1} (1-R)$（Singleton界对应$L=1$的情况，即唯一解码）进行列表解码。我们证明，对于任何固定的$L>1$，要逼近这一界需要指数级大小的字母表。具体而言，对于每个$L>1$和$R\in(0,1)$，若一个速率为$R$的码能进行列表大小为$L$、误差分数高达$\tfrac{L}{L+1} (1-R -\varepsilon)$的解码，则其字母表大小至少为$\exp(\Omega_{L,R}(1/\varepsilon))$。这与唯一解码的情况形成鲜明对比——某些速率为$R$的代数几何（AG）码族在字母表大小为$O(1/\varepsilon^2)$时，即可对误差分数高达$(1-R-\varepsilon)/2$实现唯一解码。我们的下界在指数常数因子内是紧的——使用$\exp(O_L(1/\varepsilon))$大小字母表的高概率随机码（或如近期研究所示，甚至随机线性码）能以列表大小$L$对误差分数$\tfrac{L}{L+1} (1-R -\varepsilon)$进行解码。