List Decoding of Reed–Solomon Codes

Abstract

In Chapter 6, we introduced an efficient decoder for GRS codes, yet we assumed that the number of errors does not exceed ⌊( d −1)/2⌋, where d is the minimum distance of the code. In this chapter, we present a decoding algorithm for GRS codes, due to Guruswami and Sudan, where this upper limit is relaxed. When a decoder attempts to correct more than ⌊( d −1)/2⌋ errors, the decoding may sometimes not be unique; therefore, we consider here a more general model of decoding, allowing the decoder to return a list of codewords, rather than just one codeword. In this more general setting, a decoding is considered successful if the computed list of codewords contains the transmitted codeword. The (maximum) number of errors that a list decoder can successfully handle is called the decoding radius of the decoder. The approach that leads to the Guruswami–Sudan list decoder is quite different from the GRS decoder which was introduced in Chapter 6. Specifically, the first decoding step now computes from the received word a certain bivariate polynomial Q(x, z) over the ground field, F , of the code. Regarding Q(x, z) as a univariate polynomial in the indeterminate z over the ring F[x] , a second decoding step computes the roots of Q(x, z) in F[x] ; these roots are then mapped to codewords which, in turn, form the returned list.