Interpolation-based decoding of folded variants of linearized and skew Reed–Solomon codes

Abstract

The sum-rank metric is a hybrid between the Hamming metric and the rank metric and suitable for error correction in multishot network coding and distributed storage as well as for the design of quantum-resistant cryptosystems. In this work, we consider the construction and decoding of folded linearized Reed–Solomon (FLRS) codes, which are shown to be maximum sum-rank distance (MSRD) for appropriate parameter choices. We derive an efficient interpolation-based decoding algorithm for FLRS codes that can be used as a list decoder or as a probabilistic unique decoder. The proposed decoding scheme can correct sum-rank errors beyond the unique decoding radius with a computational complexity that is quadratic in the length of the unfolded code. We show how the error-correction capability can be optimized for high-rate codes by an alternative choice of interpolation points. We derive a heuristic upper bound on the decoding failure probability of the probabilistic unique decoder and verify its tightness by Monte Carlo simulations. Further, we study the construction and decoding of folded skew Reed-Solomon codes in the skew metric. Up to our knowledge, FLRS codes are the first MSRD codes with different block sizes that come along with an efficient decoding algorithm.

Fast decoding of lifted interleaved linearized Reed–Solomon codes for multishot network coding

“…Note, that the assumption that the coefficients q (r ) i, j are uniformly distributed over F q m does not reflect the distribution of the error space tuple E. Although there is evidence that this assumption is reasonable (see e.g. [16] for folded LRS codes), it does not reflect the actual error model of the multishot operator channel.…”

Decoding of Interleaved Linearized Reed-Solomon Codes with Applications to Network Coding

“…Note, that the assumption that the coefficients q (r) i,j are uniformly distributed over F q m does not reflect the distribution of the error space tuple E. Although there is evidence that this assumption is reasonable (see e.g. [50] for folded LRS codes), it does not reflect the actual error model of the multishot operator channel. Similar as in [25,Lemma 8] for interleaved Gabidulin codes and [31,Theorem 4] for ILRS codes, the conditions of successful decoding of the interpolation-based decoder can be reduced to the conditions of the Loidreau-Overbeck-like decoder from Section 4.2.…”