Sudhir R. Ghorpade
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View article: Enumeration of minimum weight codewords of affine Cartesian codes
Enumeration of minimum weight codewords of affine Cartesian codes Open
Affine Cartesian codes were first discussed by Geil and Thomsen in 2013 in a broader framework and were formally introduced by López, Rentería-Márquez and Villarreal in 2014. These are linear error-correcting codes obtained by evaluating p…
View article: Higher weight spectra and Betti numbers of Reed-Muller codes $RM_q(2,2)$
Higher weight spectra and Betti numbers of Reed-Muller codes $RM_q(2,2)$ Open
We determine the higher weight spectra of $q$-ary Reed-Muller codes $C_q=RM_q(2,2)$ for all prime powers $q$. This is equivalent to finding the usual weight distributions of all extension codes of $C_q$ over every field extension of $F_q$ …
View article: Homotopy type of shellable $q$-complexes and their homology groups
Homotopy type of shellable $q$-complexes and their homology groups Open
The theory of shellable simplicial complexes brings together combinatorics, algebra, and topology in a remarkable way. Initially introduced by Alder for $q$-simplicial complexes, recent work of Ghorpade, Pratihar, and Randrianarisoa extend…
View article: On the minimum distance, minimum weight codewords, and the dimension of projective Reed-Muller codes
On the minimum distance, minimum weight codewords, and the dimension of projective Reed-Muller codes Open
We give an alternative proof of the formula for the minimum distance of a projective Reed-Muller code of an arbitrary order. It leads to a complete characterization of the minimum weight codewords of a projective Reed-Muller code. This is …
View article: On the Minimum Distance, Minimum Weight Codewords, and the Dimension of Projective Reed-Muller Codes
On the Minimum Distance, Minimum Weight Codewords, and the Dimension of Projective Reed-Muller Codes Open
We give an alternative proof of the formula for the minimum distance of a projective Reed-Muller code of an arbitrary order. It leads to a complete characterization of the minimum weight codewords of a projective Reed-Muller code. This is …
View article: A Combinatorial Approach to the Number of Solutions of Systems of Homogeneous Polynomial Equations over Finite Fields
A Combinatorial Approach to the Number of Solutions of Systems of Homogeneous Polynomial Equations over Finite Fields Open
We give a complete conjectural formula for the number er(d, m) of maximum possible Fq-rational points on a projective algebraic variety defined by r linearly independent homogeneous polynomial equations of degree d in m + 1 variables with …
View article: Shellability and homology of $q$-complexes and $q$-matroids
Shellability and homology of $q$-complexes and $q$-matroids Open
We consider a $q$-analogue of abstract simplicial complexes, called $q$-complexes, and discuss the notion of shellability for such complexes. It is shown that $q$-complexes formed by independent subspaces of a $q$-matroid are shellable. Fu…
View article: On the Purity of Resolutions of Stanley-Reisner Rings Associated to Reed-Muller Codes
On the Purity of Resolutions of Stanley-Reisner Rings Associated to Reed-Muller Codes Open
Following Johnsen and Verdure (2013), we can associate to any linear code $C$ an abstract simplicial complex and in turn, a Stanley-Reisner ring $R_C$. The ring $R_C$ is a standard graded algebra over a field and its projective dimension i…
View article: Arithmetic, Geometry, and Coding Theory: Homage to Gilles Lachaud
Arithmetic, Geometry, and Coding Theory: Homage to Gilles Lachaud Open
We give an overview of several of the mathematical works of Gilles Lachaud\nand provide a historical context. This is interspersed with some personal\nanecdotes highlighting many facets of his personality.\n
View article: Arithmetic, Geometry, and Coding Theory: Homage to Gilles Lachaud
Arithmetic, Geometry, and Coding Theory: Homage to Gilles Lachaud Open
We give an overview of several of the mathematical works of Gilles Lachaud and provide a historical context. This is interspersed with some personal anecdotes highlighting many facets of his personality.
View article: The role of history in learning and teaching mathematics: A personal perspective
The role of history in learning and teaching mathematics: A personal perspective Open
This is a slightly revised version of the Presidential address (General) delivered at the 84th Annual Conference of the Indian Mathematical Society held at Jammu, India during November 2018.
View article: A note on Nullstellensatz over finite fields
A note on Nullstellensatz over finite fields Open
We give an expository account of Nullstellensatz-like results when the base field is finite. In particular, we discuss the vanishing ideal of the affine space and of the projective space over a finite field. As an application, we include a…
View article: Hyperplane Sections of Determinantal Varieties over Finite Fields and\n Linear Codes
Hyperplane Sections of Determinantal Varieties over Finite Fields and\n Linear Codes Open
We determine the number of ${\\mathbb{F}}_q$-rational points of hyperplane\nsections of classical determinantal varieties defined by the vanishing of\nminors of a fixed size of a generic matrix, and identify sections giving the\nmaximum nu…
View article: A Combinatorial Approach to the Number of Solutions of Systems of Homogeneous Polynomial Equations over Finite Fields
A Combinatorial Approach to the Number of Solutions of Systems of Homogeneous Polynomial Equations over Finite Fields Open
We give a complete conjectural formula for the number $e_r(d,m)$ of maximum possible ${\mathbb{F}}q$-rational points on a projective algebraic variety defined by $r$ linearly independent homogeneous polynomial equations of degree $d$ in $m…