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

Abstract

Martínez-Peñas and Kschischang (IEEE Trans. Inf. Theory, 2019) proposed lifted linearized Reed-Solomon codes as suitable codes for error control in multishot network coding. We show how to construct and decode lifted interleaved linearized Reed-Solomon (LILRS) codes. Compared to the construction by Martínez-Peñas-Kschischang, interleaving allows to increase the decoding region significantly and decreases the overhead due to the lifting (i.e., increases the code rate), at the cost of an increased packet size. We propose two decoding schemes for LILRS that are both capable of correcting insertions and deletions beyond half the minimum distance of the code by either allowing a list or a small decoding failure probability. We propose a probabilistic unique Loidreau-Overbeck-like decoder for LILRS codes and an efficient interpolation-based decoding scheme that can be either used as a list decoder (with exponential worst-case list size) or as a probabilistic unique decoder. We derive upper bounds on the decoding failure probability of the probabilistic-unique decoders which show that the decoding failure probability is very small for most channel realizations up to the maximal decoding radius. The tightness of the bounds is verified by Monte Carlo simulations.

Error-Erasure Decoding of Linearized Reed-Solomon Codes in the Sum-Rank Metric

“…Thus, the proposed decoder can correct an overall number of insertions and deletions up to γ + δ ≤ n − k, which coincides with the decoding region of the decoders from [9], [12] and [13].…”

“…multishot network coding [2], [9], locally repairable codes [10], space-time codes [1] and code-based quantum-resistant cryptography [11]. Recently, it was shown that interleaved [12], [13] and folded [14] variants of LRS codes can be decoded beyond the unique decoding radius. The concept of row and column erasures, i.e.…”

Fast Kötter-Nielsen-Høholdt Interpolation over Skew Polynomial Rings and its Application in Coding Theory

“…We state the interpolation problem and the algorithm in a general way using linear functionals over skew polynomials rings with arbitrary automorphisms and derivations. We show how the fast KNH interpolation can be applied to interpolation-based decoding of (interleaved) Gabidulin codes [33,52], interleaved linearized Reed-Solomon codes [7,8,14,37] and (interleaved) skew Reed-Solomon codes [7,8,13].…”

“…Further applications include the construction of space-time codes [34], locally repairable codes with maximal recoverability [42] (also known as partial MDS codes) and error control for multishot network coding [41]. Interpolation-based decoding of s-interleaved LRS codes, which allows for correcting errors beyond the unique decoding radius (up to s s+1 (n − k + 1)) in the sum-rank metric, was recently considered in [7]. In the following we show, how Algorithm 3 can be used to solve the interpolation step in the interpolation-based decoder for interleaved linearized Reed-Solomon (ILRS) from [7] efficiently.…”

The non-GRS properties for the twisted generalized Reed-Solomon code and its extended code

“…An [n, k, d] linear code C over F q is a k-dimensional subspace of F n q with minimum (Hamming) distance d and length n. If the parameters reach the Singleton bound, namely, d = n − k + 1, then C is maximum distance separable (in short, MDS). Especially, generalized Reed-Solomon (in short, GRS) codes are a well known class of MDS codes, it is very important in coding theory and applications [12], [8], [18], [20], [25], [6], [17], [24], [26], [10], [1], [7], [13], [5]. Other known MDS codes have been constructed from n-arcs in projective geometry [9], circulant matrices [22], Hankel matrices [22], or twisted Reed-Solomon (in short, TGRS) codes [2], [3].…”