Convolutional Codes in Rank Metric with Application to Random Network Coding

Random network coding recently attracts attention as a technique to disseminate information in a network. This paper considers a non-coherent multi-shot network, where the unknown and time-variant network is used several times. In order to create dependencies between the different shots, particular convolutional codes in rank metric are used. These codes are so-called (partial) unit memory ((P)UM) codes, i.e., convolutional codes with memory one. First, distance measures for convolutional codes in rank metric are shown and two constructions of (P)UM codes in rank metric based on the generator matrices of maximum rank distance codes are presented. Second, an efficient error-erasure decoding algorithm for these codes is presented. Its guaranteed decoding radius is derived and its complexity is bounded. Finally, it is shown how to apply these codes for error correction in random linear and affine network coding.Comment: presented in part at Netcod 2012, submitted to IEEE Transactions on Information Theor

Skew and linearized Reed-Solomon codes and maximum sum rank distance codes over any division ring

Reed-Solomon codes and Gabidulin codes have maximum Hamming distance and maximum rank distance, respectively. A general construction using skew polynomials, called skew Reed-Solomon codes, has already been introduced in the literature. In this work, we introduce a linearized version of such codes, called linearized Reed-Solomon codes. We prove that they have maximum sum-rank distance. Such distance is of interest in multishot network coding or in singleshot multi-network coding. To prove our result, we introduce new metrics defined by skew polynomials, which we call skew metrics, we prove that skew Reed-Solomon codes have maximum skew distance, and then we translate this scenario to linearized Reed-Solomon codes and the sum-rank metric. The theories of Reed-Solomon codes and Gabidulin codes are particular cases of our theory, and the sum-rank metric extends both the Hamming and rank metrics. We develop our theory over any division ring (commutative or non-commutative field). We also consider non-zero derivations, which give new maximum rank distance codes over infinite fields not considered before

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

Codes in the sum-rank metric have various applications in error control for multishot network coding, distributed storage and code-based cryptography. Linearized Reed-Solomon (LRS) codes contain Reed-Solomon and Gabidulin codes as subclasses and fulfill the Singleton-like bound in the sum-rank metric with equality. We propose the first known error-erasure decoder for LRS codes to unleash their full potential for multishot network coding. The presented syndrome-based Berlekamp-Massey-like error-erasure decoder can correct $t_F$ full errors, $t_R$ row erasures and $t_C$ column erasures up to $2t_F + t_R + t_C \leq n-k$ in the sum-rank metric requiring at most $\mathcal{O}(n^2)$ operations in $\mathbb{F}_{q^m}$, where $n$ is the code's length and $k$ its dimension. We show how the proposed decoder can be used to correct errors in the sum-subspace metric that occur in (noncoherent) multishot network coding.Comment: 6 pages, presented at ISIT 202

Fast Decoding of Codes in the Rank, Subspace, and Sum-Rank Metric

We speed up existing decoding algorithms for three code classes in different metrics: interleaved Gabidulin codes in the rank metric, lifted interleaved Gabidulin codes in the subspace metric, and linearized Reed-Solomon codes in the sum-rank metric. The speed-ups are achieved by reducing the core of the underlying computational problems of the decoders to one common tool: computing left and right approximant bases of matrices over skew polynomial rings. To accomplish this, we describe a skew-analogue of the existing PM-Basis algorithm for matrices over usual polynomials. This captures the bulk of the work in multiplication of skew polynomials, and the complexity benefit comes from existing algorithms performing this faster than in classical quadratic complexity. The new faster algorithms for the various decoding-related computational problems are interesting in their own and have further applications, in particular parts of decoders of several other codes and foundational problems related to the remainder-evaluation of skew polynomials