Linear-Time List Decoding in Error-Free Settings

Abstract

This paper is motivated by the program of constructing list-decodable codes with linear-time encoding and decoding algorithms  with rate comparable to or even matching the rate achieved by the best constructions with polynomial encoding/decoding complexity. We achieve this for three basic settings of list decoding, and view these as the first promising steps in the above general program.

First is a setting, which we call “mixture recovering”, where for each position, the symbols of ℓ codewords are given in a scrambled order, and the goal is to recover each of the ℓ codewords. This was one of the first models studied by Ar et al in their influential paper [5] and they gave a polynomial time solution with rate 1/ℓ using Reed-Solomon codes. We propose an elegant expander-based construction with rate Ω(1/ℓ) with linear-time encoding/decoding complexity.

Second is the setting of “list-recovering” where the input is a set of ℓ possibilities for the value at each coordinate of the codeword and the goal is to find all the consistent codewords. We give an explicit linear-time encodable/decodable construction which achieves rate that is polynomial in 1/ℓ (the best rate known for polynomial decoding complexity is Ω(1/ℓ)).

Third is the setting of decoding from erasures where a certain fraction of the symbols are erased and the rest are received intact. Here, for every ε > 0, we present an explicit construction of binary codes of rate Ω(ε ² log^{− −  O(1)}(1/ε)) which can be encoded and list decoded from a fraction (1–ε) of erasures in linear time. This comes very close to the best known rate of Ω(ε ²) for polynomial decoding complexity. For codes over larger alphabets, we can even approach the optimal rate of Ω(ε) with linear time algorithms — specifically, we give linear-time list decodable codes of rate $\tilde{\Omega}(\varepsilon^{1+1/a})$ over alphabet size 2^a to recover from a fraction (1–ε) of erasures.