On the Linear Codes Arising from Schubert Varieties

Guerra, Lucio; Vincenti, Rita

doi:10.1023/b:desi.0000035470.05639.2b

Cited by 12 publications

(15 citation statements)

References 3 publications

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“…An alternative proof of the minimum distance conjecture, as well as the weight distribution of codewords in the case ℓ = 2, was obtained independently by Guerra and Vincenti [7]; in the same paper, they prove also the following lower bound for d(C α (ℓ, m)) in the general case:…”

Section: Introductionmentioning

confidence: 87%

Schubert varieties, linear codes and enumerative combinatorics

Ghorpade

Tsfasman

2005

Finite Fields and Their Applications

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We consider linear error correcting codes associated to higher dimensional projective varieties defined over a finite field. The problem of determining the basic parameters of such codes often leads to some interesting and difficult questions in combinatorics and algebraic geometry. This is illustrated by codes associated to Schubert varieties in Grassmannians, called Schubert codes, which have recently been studied. The basic parameters such as the length, dimension and minimum distance of these codes are known only in special cases. An upper bound for the minimum distance is known and it is conjectured that this bound is achieved. We give explicit formulae for the length and dimension of arbitrary Schubert codes and prove the minimum distance conjecture in the affirmative for codes associated to Schubert divisors.

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Section: Introductionmentioning

confidence: 87%

Schubert varieties, linear codes and enumerative combinatorics

Ghorpade

Tsfasman

2005

Finite Fields and Their Applications

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“…For a skew-symmetric matrix F, we know that the evaluation of F at some point W ∈ G 2,m is given by xFy T , where x and y is a basis of W . From equations (8) and 9, it is clear that if F is a skew-symmetric matrix corresponding to a codeword c A ∈ C A (2, m) then (12) c…”

Section: The Grassmannmentioning

confidence: 99%

“…This conjecture is known as the Minimum Distance Conjecture or the MDC. The MDC was proved, first by H. Chen [3] and Guerra-Vincenti [8] when = 2, then by Ghorpade-Tsfasman [7] for the Schubert divisor and finally by Xiang [19] and [6] for general Schubert codes. It is now well known that the Schubert code…”

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confidence: 99%

The weight spectrum of certain affine Grassmann codes

Piñero

Singh

2018

Des. Codes Cryptogr.

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“…This conjecture has been of some interest, and after being proved, in the affirmative, in a number of special cases (cf. [1,17,7,9]), the general case appears to have been settled very recently by Xiang [19]. The time sees ripe, therefore, to up the ante and think about more general questions.…”

Section: Introductionmentioning

confidence: 96%

Subclose families, threshold graphs, and the weight hierarchy of Grassmann and Schubert codes

Ghorpade¹,

Patil²,

Pillai³

2009

Contemporary Mathematics

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We discuss the problem of determining the complete weight hierarchy of linear error correcting codes associated to Grassmann varieties and, more generally, to Schubert varieties in Grassmannians. In geometric terms, this corresponds to determining the maximum number of Fq-rational points on sections of Schubert varieties (with nondegenerate Plücker embedding) by linear subvarieties of a fixed (co)dimension. The problem is partially solved in the case of Grassmann codes, and one of the solutions uses the combinatorial notion of a closed family. We propose a generalization of this to what is called a subclose family. A number of properties of subclose families are proved, and its connection with the notion of threshold graphs and graphs with maximum sum of squares of vertex degrees is outlined.

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On the Linear Codes Arising from Schubert Varieties

Cited by 12 publications

References 3 publications

Schubert varieties, linear codes and enumerative combinatorics

Schubert varieties, linear codes and enumerative combinatorics

The weight spectrum of certain affine Grassmann codes

Subclose families, threshold graphs, and the weight hierarchy of Grassmann and Schubert codes

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