Counting Polynomials for Linear Codes, Hyperplane Arrangements, and Matroids
Thomas-decomposition of a polynomial systems and the resulting counting polynomials are applied to the theory of linear codes, hyperplane arrangements, and vector matroids to reinterpret known polynomials such as characteristic polynomials and weight enumerator, to introduce a new polynomial counting the matrices defining the same matroid, and to introduce the concept of essential flats revealing a structure which allows to rewrite the rank generating polynomial as a sum of products of univariate polynomials. Our concepts make no essential distinction between finite and infinite fields.
2010 Mathematics Subject Classification: Primary: 05B35, 05E20, 13P99; Secondary: 05-04
Keywords and Phrases: linear codes, hyperplane arrangements, weight enumerator, vector matroid, rank generating polynomial, Thomas decomposition, counting polynomial
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