Bivariate Linear Operator Codes
Aaron L. Putterman, Vadim Zaripov
TL;DR
The paper generalizes the linear operator code framework from univariate to bivariate polynomials, formulating B-LO and its LELO extension (B-LELO) to capture a broader class of capacity-achieving, list-decodable codes. It proves a general list-decoding theorem for B-LELO codes under a list-composition criterion, unifying previous results for LO-based codes and enabling capacity-list-decodability proofs in the multivariate setting. A key achievement is showing that Permuted Product Codes (PPC) fit neatly into the B-LELO framework, thereby deriving their list-decodability up to capacity and connecting PPC with RS/FRS/Multiplicity families. The framework suggests new avenues for discovering additional capacity-achieving codes by exploring higher-dimensional operator families and their interactions, with potential impact on explicit constructions in coding theory.
Abstract
In this work, we present a generalization of the linear operator family of codes that captures more codes that achieve list decoding capacity. Linear operator (LO) codes were introduced by Bhandari, Harsha, Kumar, and Sudan [BHKS24] as a way to capture capacity-achieving codes. In their framework, a code is specified by a collection of linear operators that are applied to a message polynomial and then evaluated at a specified set of evaluation points. We generalize this idea in a way that can be applied to bivariate message polynomials, getting what we call bivariate linear operator (B-LO) codes. We show that bivariate linear operator codes capture more capacity-achieving codes, including permuted product codes introduced by Berman, Shany, and Tamo [BST24]. These codes work with bivariate message polynomials, which is why our generalization is necessary to capture them as a part of the linear operator framework. Similarly to the initial paper on linear operator codes, we present sufficient conditions for a bivariate linear operator code to be list decodable. Using this characterization, we are able to derive the theorem characterizing list-decodability of LO codes as a specific case of our theorem for B-LO codes. We also apply this theorem to show that permuted product codes are list decodable up to capacity, thereby unifying this result with those of known list-decodable LO codes, including Folded Reed-Solomon, Multiplicity, and Affine Folded Reed-Solomon codes.