arXiv subjects

Prasant Singh

Publications and source records attributed to Prasant Singh.

11 recordsLinked to original sources

arXiv2026

The Weight Spectrum of the Affine Grassmann Code $C^{\mathbb A}(3,6)$

In this article, we consider the affine Grassmann code $C^{\mathbb A}(3,6)$, obtained from the affine open cell ${\mathbb A}^9$ of the Grassmannian $G_{3,6}$. We exploit the representation of codewords as linear combinations of minors of all sizes of a generic $3\times3$ matrix and classify them according to the largest size of a minor occurring with a nonzero coefficient. Using this classification, we determine all possible Hamming weights of codewords of $C^{\mathbb A}(3,6)$ and, for each weight, compute the number of codewords attaining that weight. Consequently, we obtain the complete weight spectrum of the affine Grassmann code $C^{\mathbb A}(3,6)$.

cs.IT↗

arXiv2026

Minimum Schubert Codewords and Second-Minimum Grassmann Codewords

In this paper, we give a classification of the minimum weight codewords of Schubert codes $C_\alpha(\ell, m)$ by settling the conjecture proposed by Ghorpade and Singh in 2018, for all values of $q$ and all $\alpha$. We use this classification to prove that a codeword of the Grassmann code $C(\ell, m)$ has the second minimum weight if and only if it is indexed by an element of $\bigwedge^{m-\ell}V$ that can be written as the product of a decomposable $(m-\ell-2)$-vector and an alternating $2$-vector of rank $4$. Finally, we give an enumeration of the second minimum weight codewords of the Grassmann code.

cs.IT↗

arXiv2025

Majority Logic Decoding of Affine Grassmann Codes Over Nonbinary Fields

In this article, we consider the decoding problem of affine Grassmann codes over nonbinary fields. We use matrices of different ranks to construct a large set consisting of parity checks of affine Grassmann codes, which are orthogonal with respect to a fixed coordinate. By leveraging the automorphism groups of these codes, we generate a set of orthogonal parity checks for each coordinate. Using these parity checks, we perform majority logic decoding to correct a large number of errors in affine Grassmann codes. The order of error correction capability and the complexity of this decoder for affine Grassmann codes are the same as those of the majority logic decoder for Grassmann codes proposed in [BS21].

cs.IT↗

arXiv2024

Linear codes associated to symmetric determinantal varieties; General case

The study of linear codes over a finite field of odd cardinality, derived from determinantal varieties obtained from symmetric matrices of bounded rank, was initiated in a recent paper by the authors. There, one found the minimum distance of the code obtained from evaluating homogeneous linear functions at all symmetric matrices with rank, which is, at most, a given even number. Furthermore, a conjecture for the minimum distance of codes from symmetric matrices with ranks bounded by an odd number was given. In this article, we continue the study of codes from symmetric matrices of bounded rank. A connection between the weights of the codewords of this code and Q-numbers of the association scheme of symmetric matrices is established. Consequently, we get a concrete formula for the weight distribution of these codes. Finally, we determine the minimum distance of the code obtained from evaluating homogeneous linear functions at all symmetric matrices with rank at most a given number, both when this number is odd and when it is even.

math.AG↗

arXiv2023

Linear Codes Associated to Symmetric Determinantal Varieties: Even Rank Case

We consider linear codes over a finite field of odd characteristic, derived from determinantal varieties, obtained from symmetric matrices of bounded ranks. A formula for the weight of a code word is derived. Using this formula, we have computed the minimum distance for the codes corresponding to matrices upper-bounded by any fixed, even rank. A conjecture is proposed for the cases where the upper bound is odd. At the end of the article, tables for the weights of these codes, for spaces of symmetric matrices up to order $5$, are given. We also correct typographical errors in Proposition 1.1/3.1 of [3], and in the last table, and we have rewritten Corollary 4.9 of that paper, and the usage of that Corollary in the proof of Proposition 4.10.

cs.IT↗

arXiv2021

Orbit Structure of Grassmannian $G_{2, m}$ and a decoder for Grassmann code $C(2, m)$

In this manuscript, we consider decoding Grassmann codes, linear codes associated to Grassmannian of planes in an affine space. We look at the orbit structure of Grassmannian arising from the natural action of multiplicative group of certain finite field extension. We project the corresponding Grassmann code onto these orbits to obtain a few subcodes of certain Reed-Solomon code. We prove that some of these projected codes contains an information set of the parent Grassmann code. By improving the efficiency of Peterson's decoding algorithm for the projected subcodes, we prove that one can correct up to $\lfloor d-1/2\rfloor$ errors for Grassmann code, where $d$ is the minimum distance of Grassmann code.

cs.IT↗

arXiv2020

Pure Resolutions, Linear Codes, and Betti Numbers

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.

cs.IT↗

arXiv2020

Majority Logic Decoding for Certain Schubert Codes Using Lines in Schubert Varieties

In this article, we consider Schubert codes, linear codes associated to Schubert varieties, and discuss minimum weight codewords for dual Schubert codes. The notion of lines in Schubert varieties is looked closely at, and it has been proved that the supports of the minimum weight codewords of the dual Schubert codes lie on lines and any three points on a line in Schubert variety correspond to the support of some minimum weight parity check for the Schubert code. We use these lines in Schubert varieties to construct orthogonal parity checks for certain Schubert codes and use them for majority logic decoding. In some special cases, we can correct approximately up to $\lfloor (d-1)/2\rfloor$ many errors where $d$ is the minimum distance of the code.

cs.IT↗

arXiv2020

Point-line incidence on Grassmannians and majority logic decoding of Grassmann codes

In this article, we consider the decoding problem of Grassmann codes using majority logic. We show that for two points of the Grassmannian, there exists a canonical path between these points once a complete flag is fixed. These paths are used to construct a large set of parity checks orthogonal on a coordinate of the code, resulting in a majority decoding algorithm.

cs.IT↗

arXiv2018

Linear Codes Associated to Skew-symmetric Determinantal Varieties

In this article we consider linear codes coming from skew-symmetric determinantal varieties, which are defined by the vanishing of minors of a certain fixed size in the space of skew-symmetric matrices. In odd characteristic, the minimum distances of these codes are determined and a recursive formula for the weight of a general codeword in these codes is given.

math.CO↗

arXiv2017

Minimum Distance and the Minimum Weight Codewords of Schubert Codes

We consider linear codes associated to Schubert varieties in Grassmannians. A formula for the minimum distance of these codes was conjectured in 2000 and after having been established in various special cases, it was proved in 2008 by Xiang. We give an alternative proof of this formula. Further, we propose a characterization of the minimum weight codewords of Schubert codes by introducing the notion of Schubert decomposable elements of certain exterior powers. It is shown that codewords corresponding to Schubert decomposable elements are of minimum weight and also that the converse is true in many cases. A lower bound, and in some cases, an exact formula, for the number of minimum weight codewords of Schubert codes is also given. From a geometric point of view, these results correspond to determining the maximum number of $\mathbb{F}_q$-rational points that can lie on a hyperplane section of a Schubert variety in a Grassmannian with its nondegenerate embedding in a projective subspace of the Plücker projective space, and also the number of hyperplanes for which the maximum is attained.

cs.IT↗