Erasure List-Decodable Codes from Random and Algebraic Geometry Codes

Abstract

Erasure list decoding was introduced to correct a larger number of erasures with output of a list of possible candidates. In the present paper, we consider both random linear codes and algebraic geometry codes for list decoding erasure errors. The contributions of this paper are two-fold. Firstly, we show that, for arbitrary $0<R<1$ and $\epsilon>0$ ($R$ and $\epsilon$ are independent), with high probability a random linear code is an erasure list decodable code with constant list size $2^{O(1/\epsilon)}$ that can correct a fraction $1-R-\epsilon$ of erasures, i.e., a random linear code achieves the information-theoretic optimal trade-off between information rate and fraction of erasure errors. Secondly, we show that algebraic geometry codes are good erasure list-decodable codes. Precisely speaking, for any $0<R<1$ and $\epsilon>0$, a $q$-ary algebraic geometry code of rate $R$ from the Garcia-Stichtenoth tower can correct $1-R-\frac{1}{\sqrt{q}-1}+\frac{1}{q}-\epsilon$ fraction of erasure errors with list size $O(1/\epsilon)$. This improves the Johnson bound applied to algebraic geometry codes. Furthermore, list decoding of these algebraic geometry codes can be implemented in polynomial time.