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Journal ArticleDOI

Functional codes arising from quadric intersections with Hermitian varieties

01 Jan 2010-Finite Fields and Their Applications (Academic Press)-Vol. 16, Iss: 1, pp 27-35
TL;DR: This paper will answer the question about the minimum distance in general dimension N, with N.
About: This article is published in Finite Fields and Their Applications.The article was published on 2010-01-01 and is currently open access. It has received 19 citations till now. The article focuses on the topics: Hermitian variety & Hermitian matrix.
Citations
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Journal ArticleDOI
TL;DR: Galois geometries and coding theory are two research areas which have been interacting with each other for many decades as mentioned in this paper, from the early examples linking linear MDS codes with arcs in finite projective spaces, linear codes meeting the Griesmer bound with minihypers, covering radius with saturating sets, links have evolved to functional codes, generalized projective Reed---Muller codes and even further to LDPC codes, random network codes, and distributed storage.
Abstract: Galois geometries and coding theory are two research areas which have been interacting with each other for many decades. From the early examples linking linear MDS codes with arcs in finite projective spaces, linear codes meeting the Griesmer bound with minihypers, covering radius with saturating sets, links have evolved to functional codes, generalized projective Reed---Muller codes, and even further to LDPC codes, random network codes, and distributed storage. This article reviews briefly the known links, and then focuses on new links and new directions. We present new results and open problems to stimulate the research on Galois geometries, coding theory, and on their continuously developing and increasing interactions.

88 citations

Book Chapter
01 Jan 2011
TL;DR: The known links are reviewed, new results and open problems are presented to stimulate the research on Galois geometries, coding theory, and on their continuously developing and increasing interactions.

64 citations

Journal ArticleDOI
TL;DR: The main result of this article is that the small weight codewords correspond to the intersections of X with the singular Hermitian varieties of PG(N, q2) consisting of q + 1 hyperplanes through a common (N − 2)-dimensional space Π, forming a Baer subline in the quotient space of Π.
Abstract: This article studies the small weight codewords of the functional code C Herm (X), with X a non-singular Hermitian variety of PG(N, q 2). The main result of this article is that the small weight codewords correspond to the intersections of X with the singular Hermitian varieties of PG(N, q 2) consisting of q + 1 hyperplanes through a common (N ? 2)-dimensional space ?, forming a Baer subline in the quotient space of ?. The number of codewords having these small weights is also calculated. In this way, similar results are obtained to the functional codes C 2(Q), Q a non-singular quadric (Edoukou et al., J. Pure Appl. Algebra 214:1729---1739, 2010), and C 2(X), X a non-singular Hermitian variety (Hallez and Storme, Finite Fields Appl. 16:27---35, 2010).

17 citations

Journal ArticleDOI
TL;DR: It is proved that the small weight codewords correspond to the intersections of Q with the singular quadrics of PG(N,q) consisting of two hyperplanes.

16 citations

Dissertation
01 Jan 2014

15 citations


Cites background from "Functional codes arising from quadr..."

  • ...Its minimum distance, small weight code words and divisors have been studied before in [47, 49, 52] for n = 3, in [48] for n = 4 and in [69] for 5 ≤ n ≤ O(q(2))....

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  • ...We recall another result from [69] and we make an observation about a case that was not treated in [69]....

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References
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Book
20 Feb 1986
TL;DR: Projective Geometrics Over Finite Fields (OUP, 1979) as mentioned in this paper considers projective spaces of three dimensions over a finite field and examines properties of four and five dimensions, fundamental applications to translation planes, simple groups, and coding theory.
Abstract: This self-contained and highly detailed study considers projective spaces of three dimensions over a finite field It is the second and core volume of a three-volume treatise on finite projective spaces, the first volume being Projective Geometrics Over Finite Fields (OUP, 1979) The present work restricts itself to three dimensions, and considers both topics which are analogous of geometry over the complex numbers and topics that arise out of the modern theory of incidence structures The book also examines properties of four and five dimensions, fundamental applications to translation planes, simple groups, and coding theory

713 citations

BookDOI
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

Journal ArticleDOI
TL;DR: In this article, the authors studied the geometry of Hermitian varieties, which have properties analogous to quadrics and are defined only for finite projective spaces for which the ground (Galois field) GF(q 2) has order q 2, where q is the power of a prime.
Abstract: The geometry of quadric varieties (hypersurfaces) in finite projective spaces of N dimensions has been studied by Primrose (12) and Ray-Chaudhuri (13). In this paper we study the geometry of another class of varieties, which we call Hermitian varieties and which have many properties analogous to quadrics. Hermitian varieties are defined only for finite projective spaces for which the ground (Galois field) GF(q 2) has order q2, where q is the power of a prime. If h is any element of GF(q2), then = hq is defined to be conjugate to h.

94 citations


"Functional codes arising from quadr..." refers background in this paper

  • ...Consider a non-singular quadric or Hermitian variety X in N dimensions, then a non-tangent hyperplane intersects X in a non-singular quadric or non-singular Hermitian variety, and a tangent hyperplane intersects this non-singular quadric or Hermitian variety X in a cone π0X′ , with X′ a quadric or Hermitian variety in N − 2 dimensions of the same type as X; see [1,2] for these properties in the case of Hermitian varieties....

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Book ChapterDOI
31 Jan 1996

54 citations


"Functional codes arising from quadr..." refers background in this paper

  • ...The functional code C2(X) [10] is the linear code C2(X) = {( f (P1), ....

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  • ...Lachaud (1996) [10] in the special case where X is a non-singular Hermitian variety in PG(N,q2) and h = 2....

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  • ...…November 2009 Available online 9 December 2009 Communicated by Gary L. Mullen Keywords: Functional codes Quadrics Hermitian varieties We investigate the functional code Ch(X) introduced by G. Lachaud (1996) [10] in the special case where X is a non-singular Hermitian variety in PG(N,q2) and h = 2....

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