2,090 research outputs found
Interpolation-based Decoding of Nonlinear Maximum Rank Distance Codes
In this paper, we formulate a generic construction of MRD codes that covers almost all the newly found MRD codes. Among those MRD codes, we particularly investigate the encoding and decoding of a family of nonlinear MRD codes recently by Otal and Özbudak.acceptedVersio
Decoding and constructions of codes in rank and Hamming metric
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
A Unified Coded Deep Neural Network Training Strategy Based on Generalized PolyDot Codes for Matrix Multiplication
This paper has two contributions. First, we propose a novel coded matrix
multiplication technique called Generalized PolyDot codes that advances on
existing methods for coded matrix multiplication under storage and
communication constraints. This technique uses "garbage alignment," i.e.,
aligning computations in coded computing that are not a part of the desired
output. Generalized PolyDot codes bridge between Polynomial codes and MatDot
codes, trading off between recovery threshold and communication costs. Second,
we demonstrate that Generalized PolyDot can be used for training large Deep
Neural Networks (DNNs) on unreliable nodes prone to soft-errors. This requires
us to address three additional challenges: (i) prohibitively large overhead of
coding the weight matrices in each layer of the DNN at each iteration; (ii)
nonlinear operations during training, which are incompatible with linear
coding; and (iii) not assuming presence of an error-free master node, requiring
us to architect a fully decentralized implementation without any "single point
of failure." We allow all primary DNN training steps, namely, matrix
multiplication, nonlinear activation, Hadamard product, and update steps as
well as the encoding/decoding to be error-prone. We consider the case of
mini-batch size , as well as , leveraging coded matrix-vector
products, and matrix-matrix products respectively. The problem of DNN training
under soft-errors also motivates an interesting, probabilistic error model
under which a real number MDS code is shown to correct errors
with probability as compared to for the
more conventional, adversarial error model. We also demonstrate that our
proposed strategy can provide unbounded gains in error tolerance over a
competing replication strategy and a preliminary MDS-code-based strategy for
both these error models.Comment: Presented in part at the IEEE International Symposium on Information
Theory 2018 (Submission Date: Jan 12 2018); Currently under review at the
IEEE Transactions on Information Theor
On interpolation-based decoding of a class of maximum rank distance codes
In this paper we present an interpolation-based decoding algorithm to decode a family of maximum rank distance codes proposed recently by Trombetti and Zhou. We employ the properties of the Dickson matrix associated with a linearized polynomial with a given rank and the modified Berlekamp-Massey algorithm in decoding. When the rank of the error vector attains the unique decoding radius, the problem is converted to solving a quadratic polynomial, which ensures that the proposed decoding algorithm has polynomial-time complexity.acceptedVersio
Encoding and decoding of several optimal rank metric codes
This paper presents encoding and decoding algorithms for several families of optimal rank metric codes whose codes are in restricted forms of symmetric, alternating and Hermitian matrices. First, we show the evaluation encoding is the right choice for these codes and then we provide easily reversible encoding methods for each family. Later unique decoding algorithms for the codes are described. The decoding algorithms are interpolation-based and can uniquely correct errors for each code with rank up to ⌊(d − 1)/2⌋ in polynomial-time, where d is the minimum distance of the code.publishedVersio
On products and powers of linear codes under componentwise multiplication
In this text we develop the formalism of products and powers of linear codes
under componentwise multiplication. As an expanded version of the author's talk
at AGCT-14, focus is put mostly on basic properties and descriptive statements
that could otherwise probably not fit in a regular research paper. On the other
hand, more advanced results and applications are only quickly mentioned with
references to the literature. We also point out a few open problems.
Our presentation alternates between two points of view, which the theory
intertwines in an essential way: that of combinatorial coding, and that of
algebraic geometry.
In appendices that can be read independently, we investigate topics in
multilinear algebra over finite fields, notably we establish a criterion for a
symmetric multilinear map to admit a symmetric algorithm, or equivalently, for
a symmetric tensor to decompose as a sum of elementary symmetric tensors.Comment: 75 pages; expanded version of a talk at AGCT-14 (Luminy), to appear
in vol. 637 of Contemporary Math., AMS, Apr. 2015; v3: minor typos corrected
in the final "open questions" sectio
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