Some properties and applications of Hermitian varieties in a finite projective space PG(N, q2) in the construction of strongly regular graphs (two-class association schemes) and block designs

Chakravarti, I. M.

doi:10.1016/0095-8956(71)90037-2

Cited by 18 publications

(12 citation statements)

References 6 publications

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“…Thus, by Theorem 3.6, RM q (1, m) has a pure resolution. Consequently, the Betti numbers of RM q (1, m) can be determined using the Herzog-Kühl formula (9) as follows.…”

Section: Reed-muller Codesmentioning

confidence: 99%

Pure resolutions, linear codes, and Betti numbers

Ghorpade

Singh

2020

Journal of Pure and Applied Algebra

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We consider the minimal free resolutions of Stanley-Reisner rings associated to linear codes and give an intrinsic characterization of linear codes having a pure resolution. We use this characterization to quickly deduce the minimal free resolutions of Stanley-Reisner rings associated to MDS codes as well as constant weight codes. We also deduce that the minimal free resolutions of Stanley-Reisner rings of first order Reed-Muller codes are pure, and explicitly describe the Betti numbers. Further, we show that in the case of higher order Reed-Muller codes, the minimal free resolutions are almost always not pure. The nature of the minimal free resolution of Stanley-Reisner rings corresponding to several classes of two-weight codes, besides the first order Reed-Muller codes, is also determined.

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“…Thus, by Theorem 3.6, RM q (1, m) has a pure resolution. Consequently, the Betti numbers of RM q (1, m) can be determined using the Herzog-Kühl formula (9) as follows.…”

Section: Reed-muller Codesmentioning

confidence: 99%

Pure resolutions, linear codes, and Betti numbers

Ghorpade

Singh

2020

Journal of Pure and Applied Algebra

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“…For those who first of all, want to have a very readable treatment of the works Goppa on Hermitian curves but have a limited understanding of algebraic geometry the book of W. Cary and V. Pless [2, pp.526-544] can be examined. Some interesting results have been obtained in the case of the non-singular Hermitian surfaces by I. M. Chakravarti's group [3]. Their works were mainly done on the fields F 4 of order four.…”

Section: Introductionmentioning

confidence: 99%

Structure of functional codes defined on non-degenerate Hermitian varieties

Edoukou

Ling

Xing

2011

Journal of Combinatorial Theory, Series A

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We study the functional codes of order h defined by G. Lachaud on a non-degenerate Hermitian variety, by exhibiting a result on divisibility for all the weights of such codes. In the case where the functional code is defined by evaluating quadratic functions on the non-degenerate Hermitian surface, we list the first five weights, describe the geometrical structure of the corresponding quadrics and give a positive answer to a conjecture formulated on this question by Edoukou (2009) [8]. The paper ends with two conjectures. The first is about the divisibility of the weights in the functional codes. The second is about the minimum distance and the distribution of the codewords of the first 2h + 1 weights.

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“…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 π 0 X , 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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confidence: 99%

Functional codes arising from quadric intersections with Hermitian varieties

Hallez

Storme

2010

Finite Fields and Their Applications

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We investigate the functional code C h (X) introduced by G. Lachaud [10] in the special case where X is a non-singular Hermitian variety in PG(N, q 2) and h = 2. In [4], F. Edoukou solved the conjecture of Sørensen [11] on the minimum distance of this code for a Hermitian variety X in PG(3, q 2). In this paper, we will answer the question about the minimum distance in general dimension N , with N < O(q 2). We also prove that the small weight codewords correspond to the intersection of X with the union of 2 hyperplanes.

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Some properties and applications of Hermitian varieties in a finite projective space PG(N, q2) in the construction of strongly regular graphs (two-class association schemes) and block designs

Cited by 18 publications

References 6 publications

Pure resolutions, linear codes, and Betti numbers

Pure resolutions, linear codes, and Betti numbers

Structure of functional codes defined on non-degenerate Hermitian varieties

Functional codes arising from quadric intersections with Hermitian varieties

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