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

TL;DR

It is proved that any code exactly achieving the L-th generalized Singleton bound requires alphabet size, and any code exactly achieving the L-th generalized Singleton bound requires exponential alphabets.

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 $\exp(Ω_{L,R}(1/\varepsilon))$. 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 $2^{Ω_{L,R}(n)}$. 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 $\exp(O_L(1/\varepsilon))$-sized alphabets, can be list-of-$L$ decoded up to error fraction $\frac{L}{L+1} (1-R -\varepsilon)$.

Figures (3)

• Figure 1: The generalized Singleton bound, illustrated for $L=2$. In any code of rate $R$, by pigeonhole, there are three codewords $c_0,c_1,c_2$ that agree on the first $Rn-O(1)$ coordinates. Then there is a "list-decoding center" $y$ that differs from each of $c_0, c_1,c_2$ on at most $\frac{2}{3}(1-R)n +O(1)$ coordinates, so the code is not $(\frac{2}{3}(1-R)+o(1),2)$-list-decodable.
• Figure 2: The bad average-radius-list-decoding configuration we search for in Proposition \ref{['pr:main-1']}. The list-decoding center $y$ has distances $4\varepsilon n$, $(1-R-3\varepsilon)n$, and $(1-R-3\varepsilon)n$ from codewords $c_0$, $c_1$, and $c_2$ respectively.
• Figure 3: The agreement pattern we search for via pigeonhole in our upper bound, for $L=2$. Codeword $c_0$ differs from $y$ in at most $d_0+2d_1$ places and codewords $c_1$ and $c_2$ differ from $y$ in at most $n-d_0-d_1-a_\mathcal{F}$ places.

Theorems & Definitions (20)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3: BDG22
• Proposition 1.4
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3
• Lemma 3.4