Generalized Weight Polynomials of Codes through Flats and Orlik-Solomon Algebras of Matroids

Abstract

We present various ways of determining generalized weight polynomials of a matroid $M$, and we recall how one can find the generalized weight spectra of a linear code, given these polynomials, for the matroid determined by any generator matrix of the code. A main goal is to give coding theorists different ways to determine these polynomials. We describe how one can find the generalized weight polynomials of any matroid $M$, directly from its lattice of flats, and we also show how one can find them from the Poincare series (in this case polynomials) of the associated Orlik-Solomon algebras of matroids arising as contractions of the flats of $M$. This opens for using information about broken circuits of the matroids to determine generalized weight polynomials. We also recall the well-known connection between the Orlik-Solomon algebra of a matroid, and Whitney numbers obtained by order homology, and use it to show how one can describe weight polynomials in terms of Whitney numbers from order homology of the matroid and its contraction of flats. We recall briefly a well-known relation between the Orlik-Solomon algebra of a matroid $M$, which is representable over the complex numbers, and de Rham homology numbers obtained from complements of intersections of hyperplanes in a hyperplane arrangement corresponding to $M$. We also describe a way to find the generalized weight polynomials of a matroid in terms of polynomials defined in connection with its lattice of cyclic flats. In a simple running example we show how one can calculate the generalized weight polynomials in different ways, including both the methods presented in this paper, and selected methods developed in earlier papers. We also include a less simple example with the projective Reed-Muller code $PR_3(2,2).$