Algebraic Soft-Decision Decoding of Hermitian Codes

Lee, Kwankyu; O’Sullivan, Michael E.

doi:10.1109/tit.2010.2046208

Cited by 34 publications

(11 citation statements)

References 30 publications

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“…Later on, the research about Hermitian codes branches out in several different lines. Some papers, as for instance [21,22,25,29], deal with the problem of finding efficient algorithms for the decoding of the Hermitian codes.…”

Section: Introductionmentioning

confidence: 99%

Hermitian codes and complete intersections

Marcolla

Roggero

2020

Finite Fields and Their Applications

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In this paper we consider the Hermitian codes defined as the dual codes of one-point evaluation codes on the Hermitian curve H over the finite field F q 2 . We focus on those with distance d ≥ q 2 − q and give a geometric description of the support of their minimum-weight codewords. We consider the unique writing µq+λ(q+1) of the distance d with µ, λ non negative integers, and µ ≤ q, and consider all the curves X of the affine plane A 2 F q 2 of degree µ + λ defined by polynomials with x µ y λ as leading monomial w.r.t. the DegRevLex term ordering (with y > x). We prove that a zero-dimensional subscheme Z of A 2 F q 2 is the support of a minimum-weight codeword of the Hermitian code with distance d if and only if it is made of d simple F q 2 -points and there is a curve X such that Z coincides with the scheme theoretic intersection H ∩ X (namely, as a cycle, Z = H · X ). Finally, exploiting this geometric characterization, we propose an algorithm to compute the number of minimum weight codewords and we present comparison tables between our algorithm and MAGMA command MinimumWords.

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Section: Introductionmentioning

confidence: 99%

Hermitian codes and complete intersections

Marcolla

Roggero

2020

Finite Fields and Their Applications

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“…363-373). Information coding: the generalized Vandermonde matrix is used in coding and decoding information in the Hermitian code [16]. Optimization of the non-homogeneous differential equation [17].…”

Section: Practical Importance Of the Generalized Vandermonde Matrixmentioning

confidence: 99%

Recursive Matrix Calculation Paradigm by the Example of Structured Matrix

Respondek

2020

Information

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In this paper, we derive recursive algorithms for calculating the determinant and inverse of the generalized Vandermonde matrix. The main advantage of the recursive algorithms is the fact that the computational complexity of the presented algorithm is better than calculating the determinant and the inverse by means of classical methods, developed for the general matrices. The results of this article do not require any symbolic calculations and, therefore, can be performed by a numerical algorithm implemented in a specialized (like Matlab or Mathematica) or general-purpose programming language (C, C++, Java, Pascal, Fortran, etc.).

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“…As is well known, the low-rank Hermitian matrix approximation problems occur in many areas, specially in finance, and have been widely researched, see [11,12,9,13]. Some works about the matrices leading to the Hermitian solutions have been given in [14][15][16] and some useful algorithms have also been discussed in those papers. On the other hand, the solution of Hermitian-type matrix equation A = BXB H has also been investigated.…”

Section: Introductionmentioning

confidence: 99%

Minimum rank (skew) Hermitian solutions to the matrix approximation problem in the spectral norm

Shen

Wei

Liu

2015

Journal of Computational and Applied Mathematics

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MSC: 15A24 15A60 93A10Keywords: Matrix approximation Minimum rank Norm-preserving dilations HGSVD SHGSVD a b s t r a c tIn this paper, we discuss the following two minimum rank matrix approximation problems in the spectral norm:By applying the norm-preserving dilation theorem, the Hermitian-type (skewHermitian-type) generalized singular value decomposition (HGSVD, SHGSVD), we characterize the expressions of the minimum rank and derive a general form of minimum rank (skew) Hermitian solutions to the matrix approximation problem.

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Algebraic Soft-Decision Decoding of Hermitian Codes

Cited by 34 publications

References 30 publications

Hermitian codes and complete intersections

Hermitian codes and complete intersections

Recursive Matrix Calculation Paradigm by the Example of Structured Matrix

Minimum rank (skew) Hermitian solutions to the matrix approximation problem in the spectral norm

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