Line Hermitian Grassmann codes and their parameters
TL;DR: These subcodes of the Grassmann codes associated to the $2$-Grassmannian of a Hermitian polar space defined over a finite field of square order are introduced and studied.
About: This article is published in Finite Fields and Their Applications.The article was published on 2018-05-01 and is currently open access. It has received 4 citations till now. The article focuses on the topics: Hermitian variety & Hermitian matrix.
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TL;DR: In this paper, an algorithm for the point enumerator of a line Hermitian Grassmannian code is presented, which can be usefully applied to get efficient encoders, decoders and error correction algorithms for the aforementioned codes.
1 citations
23 Oct 2022
TL;DR: It is proved that the minimum distance of the polar orthogonal Grassmann code C ( O 3 , 6 ) is q 3 − q 3 for q odd and q 3For q even.
Abstract: The polar orthogonal Grassmann code C ( O 3 , 6 ) is the linear code associated to the Grassmann embedding of the Dual Polar space of Q + (5 , q ). In this manuscript we study the minimum distance of this embedding. We prove that the minimum distance of the polar orthogonal Grassmann code C ( O 3 , 6 ) is q 3 − q 3 for q odd and q 3 for q even. Our technique is based on partitioning the orthogonal space into different sets such that on each partition the code C ( O 3 , 6 ) is identified with evaluations of determinants of skew–symmetric matrices. Our bounds come from elementary algebraic methods counting the zeroes of particular classes of polynomials. We expect our techniques may be applied to other polar Grassmann codes.
TL;DR: A new class of linear codes, called affine symplectic Grassmann codes, are introduced, and their parameters, automorphism group, minimum distance codewords, dual code and other key features are determined.
Abstract: In this manuscript, we introduce a new class of linear codes, called affine symplectic Grassmann codes, and determine their parameters, automorphism group, minimum distance codewords, dual code and other key features. These linear codes are defined from an affine part of a polar symplectic Grassmannian. They combine polar symplectic Grassmann codes and affine Grassmann codes.
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TL;DR: Affine Hermitian Grassmann codes as discussed by the authors are the linear codes resulting from an affine part of the projection of the Polar Hermitians of the Grassmannian codes.
Abstract: The Grassmannian is an important object in Algebraic Geometry. One of the many techniques used to study the Grassmannian is to build a vector space from its points in the projective embedding and study the properties of the resulting linear code.
We introduce a new class of linear codes, called Affine Hermitian Grassman Codes. These codes are the linear codes resulting from an affine part of the projection of the Polar Hermitian Grassmann codes. They combine Polar Hermitian Grassmann codes and Affine Grassmann codes. We will determine the parameters of these codes.
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01 Jan 2016
TL;DR: In this paper, the authors define Hermitian varieties, Grassmann varieties, Veronese and Segre varieties, and embedded geometries for finite projective spaces of three dimensions.
Abstract: Terminology Quadrics Hermitian varieties Grassmann varieties Veronese and Segre varieties Embedded geometries Arcs and caps Appendix VI. Ovoids and spreads of finite classical polar spaces Appendix VII. Errata for Finite projective spaces of three dimensions and Projective geometries over finite fields Bibliography Index of notation Author index General index.
647 citations
TL;DR: This work provides an explicit scheme for calculating the index of any sequence in S according to its position in the lexicographic ordering of S, thus resulting in a data compression of (log\midS\mid)/n.
Abstract: Let S be a given subset of binary n-sequences. We provide an explicit scheme for calculating the index of any sequence in S according to its position in the lexicographic ordering of S . A simple inverse algorithm is also given. Particularly nice formulas arise when S is the set of all n -sequences of weight k and also when S is the set of all sequences having a given empirical Markov property. Schalkwijk and Lynch have investigated the former case. The envisioned use of this indexing scheme is to transmit or store the index rather than the sequence, thus resulting in a data compression of (\log\midS\mid)/n .
565 citations
"Line Hermitian Grassmann codes and ..." refers methods in this paper
...For both these families we also proposed in [6] an efficient encoding algorithm, based on the techniques of enumerative coding introduced in [12]....
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Book•
01 Sep 2007TL;DR: The theory of algebraic geometric codes, a subject formed on the border of several domains of mathematics, is studied in this paper, where the authors constantly look for interpretations that connect coding theory to algebraic geometry and number theory.
Abstract: The book is devoted to the theory of algebraic geometric codes, a subject formed on the border of several domains of mathematics On one side there are such classical areas as algebraic geometry and number theory; on the other, information transmission theory, combinatorics, finite geometries, dense packings, etc The authors give a unique perspective on the subject Whereas most books on coding theory build up coding theory from within, starting from elementary concepts and almost always finishing without reaching a certain depth, this book constantly looks for interpretations that connect coding theory to algebraic geometry and number theory There are no prerequisites other than a standard algebra graduate course The first two chapters of the book can serve as an introduction to coding theory and algebraic geometry respectively Special attention is given to the geometry of curves over finite fields in the third chapter Finally, in the last chapter the authors explain relations between all of these: the theory of algebraic geometric codes
314 citations
237 citations
"Line Hermitian Grassmann codes and ..." refers background in this paper
...By Chow’s theorem [11], the semilinear automorphism group stabilizing the variety G(m, k) is the projective general semilinear group PΓL(m,K) unless k = m/2, in which case it is PΓL(m,K) ⋊ Z2....
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TL;DR: In this paper, considerazioni preliminari (nn.1.1-6). II: Antipolarita in spazi pascaliani, and matrici antiortogonali in spasi lineari giacenti.
Abstract: I: Considerazioni preliminari (nn.1–6). II: Antipolarita in spazi pascaliani (nn.7–15). III: Matrici hermitiane e matrici antiortogonali (nn.16–21). IV: Prime proprieta ulteriori delle suddette matrici nel caso finito (nn.22–28). V: Forme H hermitiane in spazi di Galois e spazi lineari giacenti su di esse (nn.29–33). VI: Automorfismi e antiomorfismi di una H in se (nn.34–47). VII: La geometria definita da una forma hermitiana (nn.48–61). VIII: Sul piano inversivo avente come cerchi i gruppi hermitiani di una retta di Galois (nn.62–69). IX: Forme hermitiane e quadriche permutabili (nn.70–87). X: Ricoprimenti di forme hermitiane mediante sistemi regolari di spazi, ed emisistemi (nn.88–103). XI: Risoluzione di problemi di Steiner e di Kirkman con l'uso di curve hermitiane, e questioni collegate (nn.104–109).
221 citations
"Line Hermitian Grassmann codes and ..." refers methods in this paper
...use, we refer to the monograph [23] as well as well as to the survey [2]; see also [17, Chapter 2]....
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