The weight spectrum of certain affine Grassmann codes

Fernando Piñero*, Prasant Singh

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review

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Abstract

We consider the linear code corresponding to a special affine part of the Grassmannian G2 , m, which we denote by CA(2 , m). This affine part is the complement of the Schubert divisor of G2 , m. In view of this, we show that there is a projection of Grassmann code onto the affine Grassmann code which is also a linear isomorphism. This implies that the dimensions of Grassmann codes and affine Grassmann codes are equal. The projection gives a 1–1 correspondence between codewords of Grassmann codes and affine Grassmann codes. Using this isomorphism and the correspondence between codewords, we give a skew–symmetric matrix in some standard block form corresponding to every codeword of CA(2 , m). The weight of a codeword is given in terms of the rank of some blocks of this form and it is shown that the weight of every codeword is divisible by some power of q. We also count the number of skew–symmetric matrices in the block form to compute the weight spectrum of the affine Grassmann code CA(2 , m).

Original languageEnglish
JournalDesigns, Codes, and Cryptography
Volume87
Issue number4
Pages (from-to)817-830
Number of pages14
ISSN0925-1022
DOIs
Publication statusPublished - 2019

Keywords

  • Affine Grassmannian
  • Grassmann varieties
  • Linear codes
  • Schubert varieties
  • Weight distribution

Cite this

Piñero, Fernando ; Singh, Prasant. / The weight spectrum of certain affine Grassmann codes. In: Designs, Codes, and Cryptography. 2019 ; Vol. 87, No. 4. pp. 817-830.
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The weight spectrum of certain affine Grassmann codes. / Piñero, Fernando; Singh, Prasant.

In: Designs, Codes, and Cryptography, Vol. 87, No. 4, 2019, p. 817-830.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - The weight spectrum of certain affine Grassmann codes

AU - Piñero, Fernando

AU - Singh, Prasant

PY - 2019

Y1 - 2019

N2 - We consider the linear code corresponding to a special affine part of the Grassmannian G2 , m, which we denote by CA(2 , m). This affine part is the complement of the Schubert divisor of G2 , m. In view of this, we show that there is a projection of Grassmann code onto the affine Grassmann code which is also a linear isomorphism. This implies that the dimensions of Grassmann codes and affine Grassmann codes are equal. The projection gives a 1–1 correspondence between codewords of Grassmann codes and affine Grassmann codes. Using this isomorphism and the correspondence between codewords, we give a skew–symmetric matrix in some standard block form corresponding to every codeword of CA(2 , m). The weight of a codeword is given in terms of the rank of some blocks of this form and it is shown that the weight of every codeword is divisible by some power of q. We also count the number of skew–symmetric matrices in the block form to compute the weight spectrum of the affine Grassmann code CA(2 , m).

AB - We consider the linear code corresponding to a special affine part of the Grassmannian G2 , m, which we denote by CA(2 , m). This affine part is the complement of the Schubert divisor of G2 , m. In view of this, we show that there is a projection of Grassmann code onto the affine Grassmann code which is also a linear isomorphism. This implies that the dimensions of Grassmann codes and affine Grassmann codes are equal. The projection gives a 1–1 correspondence between codewords of Grassmann codes and affine Grassmann codes. Using this isomorphism and the correspondence between codewords, we give a skew–symmetric matrix in some standard block form corresponding to every codeword of CA(2 , m). The weight of a codeword is given in terms of the rank of some blocks of this form and it is shown that the weight of every codeword is divisible by some power of q. We also count the number of skew–symmetric matrices in the block form to compute the weight spectrum of the affine Grassmann code CA(2 , m).

KW - Affine Grassmannian

KW - Grassmann varieties

KW - Linear codes

KW - Schubert varieties

KW - Weight distribution

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DO - 10.1007/s10623-018-0567-1

M3 - Journal article

VL - 87

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SN - 0925-1022

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