Explicit Subcodes of Reed-Solomon Codes that Efficiently Achieve List Decoding Capacity
Amit Berman, Yaron Shany, Itzhak Tamo
TL;DR
This work constructs explicit large subcodes of Reed–Solomon codes that achieve list-decoding capacity with a constant output list size. The authors introduce permuted product codes built from the tensor product of two RS codes and two coprime affine maps, yielding length $n\approx q-1$ and alphabet size $q^{O(1/\varepsilon^3)}$ while decoding up to a fraction $1-R-\varepsilon$ of errors with list size $L=(1/\varepsilon)^{O(1/\varepsilon^2)}$ in polynomial time. An alternative realization via bivariate polynomials on orbits of two affine transformations extends length to the field size without requiring prime fields. The approach provides a simple, explicit construction (without subspace designs) that achieves capacity with efficient decoding, contributing a new method for capacity-achieving subcodes of RS codes and opening questions about multi-affine generalizations and encoding efficiency.
Abstract
In this paper, we introduce a novel explicit family of subcodes of Reed-Solomon (RS) codes that efficiently achieve list decoding capacity with a constant output list size. Our approach builds upon the idea of large linear subcodes of RS codes evaluated on a subfield, similar to the method employed by Guruswami and Xing (STOC 2013). However, our approach diverges by leveraging the idea of {\it permuted product codes}, thereby simplifying the construction by avoiding the need of {\it subspace designs}. Specifically, the codes are constructed by initially forming the tensor product of two RS codes with carefully selected evaluation sets, followed by specific cyclic shifts to the codeword rows. This process results in each codeword column being treated as an individual coordinate, reminiscent of prior capacity-achieving codes, such as folded RS codes and univariate multiplicity codes. This construction is easily shown to be a subcode of an interleaved RS code, equivalently, an RS code evaluated on a subfield. Alternatively, the codes can be constructed by the evaluation of bivariate polynomials over orbits generated by \emph{two} affine transformations with coprime orders, extending the earlier use of a single affine transformation in folded RS codes and the recent affine folded RS codes introduced by Bhandari {\it et al.} (IEEE T-IT, Feb.~2024). While our codes require large, yet constant characteristic, the two affine transformations facilitate achieving code length equal to the field size, without the restriction of the field being prime, contrasting with univariate multiplicity codes.