• Corpus ID: 231740907

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

  title={On the Purity of Resolutions of Stanley-Reisner Rings Associated to Reed-Muller Codes},
  author={Sudhir R. Ghorpade and Rati Ludhani},
Following Johnsen and Verdure (2013), we can associate to any linear code C an abstract simplicial complex and in turn, a Stanley-Reisner ring RC . The ring RC is a standard graded algebra over a field and its projective dimension is precisely the dimension of C. Thus RC admits a graded minimal free resolution and the resulting graded Betti numbers are known to determine the generalized Hamming weights of C. The question of purity of the minimal free resolution of RC was considered by Ghorpade… 

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.



On Generalized Reed-Muller Codes and Their Relatives

A generalization of weight polynomials to matroids

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

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.

Generalized Hamming weights of affine Cartesian codes

Generalized Hamming weights of q-ary Reed-Muller codes

The order bound on generalized Hamming weights is introduced in a general setting of codes on varieties which comprises both the geometric Goppa codes as the q-ary Reed-Muller codes. It is shown that

Extended and Generalized Weight Enumerators

This paper gives a survey on extended and generalized weight enumerators of a linear code and the Tutte polynomial of the matroid of the code (16). Furthermore ongoing research is reported on the

A note on Nullstellensatz over finite fields

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

Application of Boolean algebra to switching circuit design and to error detection

  • D. E. Muller
  • Computer Science
    Trans. I R E Prof. Group Electron. Comput.
  • 1954
It is shown that certain parts of the multiple output problem for switching circuits that have more than one output may be reduced to a single output problem whose inputs are equal in number to the sum of the numbers of inputs and outputs in the original problem.