Pure Resolutions, Linear Codes, and Betti Numbers

  title={Pure Resolutions, Linear Codes, and Betti Numbers},
  author={Sudhir R. Ghorpade and Prasant Singh},

Free Resolutions and Generalized Hamming Weights of binary linear codes

It is proved that the first and second generalized Hamming weights of a binary linear code can be computed from a set of monomials associated with a binomial ideal related with the code.

Generalized weights of codes over rings and invariants of monomial ideals

An algebraic theory of supports for R-linear codes of fixed length, where R is a finite commutative unitary ring, states that the generalized weights of a code can be obtained from the graded Betti numbers of its associated monomial ideal.

Möbius and coboundary polynomials for matroids

It is explained how the connection with these Stanley–Reisner rings forces the coefficients of the two-variable coboundary polynomials and Möbius polynomers to satisfy certain universal equations.

On the Purity of Resolutions of Stanley-Reisner Rings Associated to Reed-Muller Codes

A complete characterization of the purity of graded minimal free resolutions of StanleyReisner rings associated to generalized Reed-Muller codes of an arbitrary order is given.



A generalization of weight polynomials to matroids

Graded Betti numbers of Cohen–Macaulay modules and the multiplicity conjecture

We give conjectures on the possible graded Betti numbers of Cohen–Macaulay modules up to multiplication by positive rational numbers. The idea is that the Betti diagrams should be non‐negative linear

The Geometry of Two‐Weight Codes

On etudie les relations entre les codes [n,k] lineaires a deux poids, les ensembles (n,k,h 1 h 2 ) projectifs et certains graphes fortement reguliers

Hermitian Varieties in a Finite Projective Space PG(N, q 2)

The geometry of quadric varieties (hypersurfaces) in finite projective spaces of N dimensions has been studied by Primrose (12) and Ray-Chaudhuri (13). In this paper we study the geometry of another


Hamming weights and Betti numbers of Stanley–Reisner rings associated to matroids

This work shows how the weights of a matroid M are determined by the Stanley–Reisner ring of the simplicial complex whose faces are the independent sets of $$M$$, and derives some consequences.

Cohen-Macaulay Complexes

Let Δ be a finite simplicial complex (or complex for short) on the vertex set V = (x1,…,xn). Thus, Δ is a collection of subsets of V satisfying the two conditions: (i) (xi) e Δ for all xi e V, and

Some maximal arcs in finite projective planes

On the bettinumbers of finite pure and linear resolutions

A characterization in terms of the Bettinumbers for a module possessing a pure resolution to be Cohen-Macauiay is given, the conjecture that the Bettinumbers should satisfy is being proven roi the