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

  title={Decoding of Interleaved Linearized Reed-Solomon Codes with Applications to Network Coding},
  author={Hannes Bartz and Sven Puchinger},
  journal={2021 IEEE International Symposium on Information Theory (ISIT)},
  • H. BartzS. Puchinger
  • Published 14 January 2021
  • Computer Science
  • 2021 IEEE International Symposium on Information Theory (ISIT)
Recently, Martínez-Peñas and Kschischang (IEEE Trans. Inf. Theory, 2019) showed that lifted linearized Reed-Solomon codes are suitable codes for error control in multishot network coding. We show how to construct and decode lifted interleaved linearized Reed-Solomon codes. Compared to the construction by Martínez-Peñas-Kschischang, interleaving allows to increase the decoding region significantly (especially w.r.t. the number of insertions) and decreases the overhead due to the lifting (i.e… 

Figures and Tables from this paper

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

It is proved that almost all ofTGRS codes and extended TGRS codes are non-GRS when the code rate more than one half.

Fast Kötter-Nielsen-Høholdt Interpolation over Skew Polynomial Rings

A fast divide-and-conquer variant of K¨otter–Nielsen–Høholdt (KNH) interpolation algorithm that inputs a list of linear functionals on skew polynomial vectors, and outputs a reduced Gr¨obner basis of their kernel intersection and matches the previous best speeds for these tasks.

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

This work proposes the first known error-erasure decoder for LRS codes to unleash their full potential for multishot network coding by incorporating erasures into the known syndrome-based Berlekamp-Massey-like decoder.

Optimal Anticodes, MSRD Codes, and Generalized Weights in the Sum-Rank Metric

It is proved that the generalized weights of an MSRD code are determined by its parameters, which explains how generalized weights measure information leakage in multishot network coding.

Efficient Decoding of Folded Linearized Reed-Solomon Codes in the Sum-Rank Metric

This work shows how to construct h -folded linearized Reed–Solomon (FLRS) codes and derive an interpolation-based decoding scheme that is capable of correcting sum-rank errors beyond the unique decoding radius and derives a heuristic upper bound on the failure probability of the probabilistic unique decoder.



Reliable and Secure Multishot Network Coding using Linearized Reed-Solomon Codes

A Welch-Berlekamp sum-rank decoding algorithm for linearized Reed-Solomon codes is provided, having quadratic complexity in the total length, and which can be adapted to handle not only errors, but also erasures, wire-tap observations and non-coherent communication.

Decoding For Iterative Reed-solomon Coding Schemes

A novel approach for the decoding of interleaved codes is proposed based on a notion of covering error locators that locate error positions in two or more adjacent words that can correct more errors than the designed error correction ability.

Multishot codes for network coding: Bounds and a multilevel construction

  • R. NóbregaB. Filho
  • Computer Science
    2009 IEEE International Symposium on Information Theory
  • 2009
This paper explores the idea of using the subspace channel more than once and investigates the so called multishot subspace codes, and presents definitions for the problem, a motivating example, lower and upper bounds for the size of Codes, and a multilevel construction of codes based on block-coded modulation.

Residues of skew rational functions and linearized Goppa codes

  • X. Caruso
  • Computer Science, Mathematics
  • 2019
The main objective is to develop a theory of residues for skew rational functions (which are, by definition, the quotients of two skew polynomials) and prove a skew analogue of the residue formula and a skew analog of the classical formula of change of variables for residues.

Coding for Errors and Erasures in Random Network Coding

A Reed-Solomon-like code construction, related to Gabidulin's construction of maximum rank-distance codes, is described and a Sudan-style ldquolist-1rdquo minimum-distance decoding algorithm is provided.

Efficient decoding of interleaved subspace and Gabidulin codes beyond their unique decoding radius using Gröbner bases

An interpolation-based decoding scheme for L-interleaved subspace codes is presented and a complementary decoding approach is presented which corrects γ insertions and δ deletions up to Lγ +δ ≤ L(nt −k).

Kötter interpolation in skew polynomial rings

This work applies Kötter’s interpolation framework to free modules over skew polynomial rings and introduces a simple interpolation algorithm akin to Newton interpolation for ordinary polynomials.

Bounds on collaborative decoding of interleaved Hermitian codes and virtual extension

  • Sabine Kampf
  • Computer Science
    Designs, Codes and Cryptography
  • 2012
A decoding algorithm that achieves the maximum decoding radius for interleaved Hermitian (IH) codes if a collaborative decoding scheme is used is given and a bound on the code rate below which virtual extension of non-interleaved hermitian codes can improve the decoding capabilities is derived.

Fundamental Properties of Sum-Rank-Metric Codes

This paper investigates the theory of sum-rank-metric codes for which the individual matrix blocks may have different sizes. Various bounds on the cardinality of a code are derived, along with their

Sum-Rank BCH Codes and Cyclic-Skew-Cyclic Codes

Tables are provided showing that sum-rank BCH codes beat previously known codes for the sum-Rank metric for binary matrices (i.e., codes whose codewords are lists of lists of Binary matrices, for a wide range of list lengths that correspond to the code length).