49 research outputs found

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    New Explicit Good Linear Sum-Rank-Metric Codes

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    Sum-rank-metric codes have wide applications in universal error correction and security in multishot network, space-time coding and construction of partial-MDS codes for repair in distributed storage. Fundamental properties of sum-rank-metric codes have been studied and some explicit or probabilistic constructions of good sum-rank-metric codes have been proposed. In this paper we propose three simple constructions of explicit linear sum-rank-metric codes. In finite length regime, numerous good linear sum-rank-metric codes from our construction are given. Most of them have better parameters than previous constructed sum-rank-metric codes. For example a lot of small block size better linear sum-rank-metric codes over Fq{\bf F}_q of the matrix size 2×22 \times 2 are constructed for q=2,3,4q=2, 3, 4. Asymptotically our constructed sum-rank-metric codes are closing to the Gilbert-Varshamov-like bound on sum-rank-metric codes for some parameters. Finally we construct a linear MSRD code over an arbitrary finite field Fq{\bf F}_q with various matrix sizes n1>n2>>ntn_1>n_2>\cdots>n_t satisfying nini+12++nt2n_i \geq n_{i+1}^2+\cdots+n_t^2 , i=1,2,,t1i=1, 2, \ldots, t-1, for any given minimum sum-rank distance. There is no restriction on the block lengths tt and parameters N=n1++ntN=n_1+\cdots+n_t of these linear MSRD codes from the sizes of the fields Fq{\bf F}_q.Comment: 32 pages, revised version, merged with arXiv:2206.0233

    Non-minimum tensor rank Gabidulin codes

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    The tensor rank of some Gabidulin codes of small dimension is investigated. In particular, we determine the tensor rank of any rank metric code equivalent to an 8-dimensional Fq-linear generalized Gabidulin code in Fq4×4. This shows that such a code is never minimum tensor rank. In this way, we detect the first infinite family of Gabidulin codes which are not minimum tensor rank

    Decoding and constructions of codes in rank and Hamming metric

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    As coding theory plays an important role in data transmission, decoding algorithms for new families of error correction codes are of great interest. This dissertation is dedicated to the decoding algorithms for new families of maximum rank distance (MRD) codes including additive generalized twisted Gabidulin (AGTG) codes and Trombetti-Zhou (TZ) codes, decoding algorithm for Gabidulin codes beyond half the minimum distance and also encoding and decoding algorithms for some new optimal rank metric codes with restrictions. We propose an interpolation-based decoding algorithm to decode AGTG codes where the decoding problem is reduced to the problem of solving a projective polynomial equation of the form q(x) = xqu+1 +bx+a = 0 for a,b ∈ Fqm. We investigate the zeros of q(x) when gcd(u,m)=1 and proposed a deterministic algorithm to solve a linearized polynomial equation which has a close connection to the zeros of q(x). An efficient polynomial-time decoding algorithm is proposed for TZ codes. The interpolation-based decoding approach transforms the decoding problem of TZ codes to the problem of solving a quadratic polynomial equation. Two new communication models are defined and using our models we manage to decode Gabidulin codes beyond half the minimum distance by one unit. Our models also allow us to improve the complexity for decoding GTG and AGTG codes. Besides working on MRD codes, we also work on restricted optimal rank metric codes including symmetric, alternating and Hermitian rank metric codes. Both encoding and decoding algorithms for these optimal families are proposed. In all the decoding algorithms presented in this thesis, the properties of Dickson matrix and the BM algorithm play crucial roles. We also touch two problems in Hamming metric. For the first problem, some cryptographic properties of Welch permutation polynomial are investigated and we use these properties to determine the weight distribution of a binary linear codes with few weights. For the second one, we introduce two new subfamilies for maximum weight spectrum codes with respect to their weight distribution and then we investigate their properties.Doktorgradsavhandlin

    New classes of nonassociative divison algebras and MRD codes

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    In the first part of the thesis, we generalize a construction by J Sheekey that employs skew polynomials to obtain new nonassociative division algebras and maximum rank distance (MRD) codes. This construction contains Albert’s twisted fields as special cases. As a byproduct, we obtain a class of nonassociative real division algebras of dimension four which has not been described in the literature so far in this form. We also obtain new MRD codes. In the second part of the thesis, we study a general doubling process (similar to the one that can be used to construct the complex numbers from pairs of real numbers) to obtain new non-unital nonassociative algebras, starting with cyclic algebras. We investigate the automorphism groups of these algebras and when they are division algebras. In particular, we obtain a generalization of Dickson’s commutative semifields. We are using methods from nonassociative algebra throughout

    Systematic maximum sum rank codes

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    In the last decade there has been a great interest in extending results for codes equipped with the Hamming metric to analogous results for codes endowed with the rank metric. This work follows this thread of research and studies the characterization of systematic generator matrices (encoders) of codes with maximum rank distance. In the context of Hamming distance these codes are the so-called Maximum Distance Separable (MDS) codes and systematic encoders have been fully investigated. In this paper we investigate the algebraic properties and representation of encoders in systematic form of Maximum Rank Distance (MRD) codes and Maximum Sum Rank Distance (MSRD) codes. We address both block codes and convolutional codes separately and present necessary and sufficient conditions for an encoder in systematic form to generate a code with maximum (sum) rank distance. These characterizations are given in terms of certain matrices that must be superregular in a extension field and that preserve superregularity after some transformations performed over the base field. We conclude the work presenting some examples of Maximum Sum Rank convolutional codes over small fields. For the given parameters the examples obtained are over smaller fields than the examples obtained by other authors.publishe

    Fundamental Properties of Sum-Rank Metric Codes

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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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