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Register a new userFundamental Properties of Sum-Rank Metric Codes
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Added by
Alberto Ravagnani
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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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 asymptotic extensions. The duality theory of sum-rank metric codes is also explored, showing that MSRD codes (the sum-rank analogue of MDS codes) dualize to MSRD codes only if all matrix blocks have the same number of columns. In the latter case, duality considerations lead to an upper bound on the number of blocks for MSRD codes. The paper also contains various constructions of sum-rank metric codes for variable block sizes, illustrating the possible behaviours of these objects with respect to bounds, existence, and duality properties.
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We derive simplified sphere-packing and Gilbert--Varshamov bounds for codes in the sum-rank metric, which can be computed more efficiently than previous ones. They give rise to asymptotic bounds that cover the asymptotic setting that has not yet been considered in the literature: families of sum-rank-metric codes whose block size grows in the code length. We also provide two genericity results: we show that random linear codes achieve almost the sum-rank-metric Gilbert--Varshamov bound with high probability. Furthermore, we derive bounds on the probability that a random linear code attains the sum-rank-metric Singleton bound, showing that for large enough extension fields, almost all linear codes achieve it.
This paper extends the study of rank-metric codes in extension fields $mathbb{L}$ equipped with an arbitrary Galois group $G = mathrm{Gal}(mathbb{L}/mathbb{K})$. We propose a framework for studying these codes as subspaces of the group algebra $mathbb{L}[G]$, and we relate this point of view with usual notions of rank-metric codes in $mathbb{L}^N$ or in $mathbb{K}^{Ntimes N}$, where $N = [mathbb{L} : mathbb{K}]$. We then adapt the notion of error-correcting pairs to this context, in order to provide a non-trivial decoding algorithm for these codes. We then focus on the case where $G$ is abelian, which leads us to see codewords as elements of a multivariate skew polynomial ring. We prove that we can bound the dimension of the vector space of zeroes of these polynomials, depending of their degree. This result can be seen as an analogue of Alon-Furedi theorem -- and by means, of Schwartz-Zippel lemma -- in the rank metric. Finally, we construct the counterparts of Reed-Muller codes in the rank metric, and we give their parameters. We also show the connection between these codes and classical Reed-Muller codes in the case where $mathbb{L}$ is a Kummer extension.
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 new algorithms that reduce the cores 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 ordinary 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 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.
In analogy with the Singleton defect for classical codes, we propose a definition of rank defect for Delsarte rank-metric codes. We characterize codes whose rank defect and dual rank defect are both zero, and prove that the rank distribution of such codes is determined by their parameters. This extends a result by Delsarte on the rank distribution of MRD codes. In the general case of codes of positive defect, we show that the rank distribution is determined by the parameters of the code, together the number of codewords of small rank. Moreover, we prove that if the rank defect of a code and its dual are both one, and the dimension satisfies a divisibility condition, then the number of minimum-rank codewords and dual minimum-rank codewords is the same. Finally, we discuss how our results specialize to Gabidulin codes.
In this paper motivated from subspace coding we introduce subspace-metric and subset-metric codes. These are coordinate-position independent pseudometrics and suitable for the folded codes introduced by Guruswami and Rudra. The half-Singleton upper bounds for linear subspace-metric and subset-metric codes are proved. Subspace distances and subset distances of codes are natural lower bounds for insdel distances of codes, and then can be used to lower bound the insertion-deletion error-correcting capabilities of codes. The problem to construct efficient insertion-deletion error-correcting codes is notorious difficult and has attracted a long-time continuous efforts. The recent breakthrough is the algorithmic construction of near-Singleton optimal rate-distance tradeoff insertion-deletion code families by B. Haeupler and A. Shahrasbi in 2017 from their synchronization string technique. However most nice codes in these recent results are not explicit though many of them can be constructed by highly efficient algorithms. Our subspace-metric and subset-metric codes can be used to construct systemic explicit well-structured insertion-deletion codes. We present some near-optimal subspace-metric and subset-metric codes from known constant dimension subspace codes. By analysing the subset distances of folded codes from evaluation codes of linear mappings, we prove that they have high subset distances and then are explicit good insertion-deletion codes
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