Hamming polynomial of a demimatroid

Abstract

Johnsen, Roksvold and Verdure have shown how fundamental invariants and concepts of linear coding theory, such as weight enumerators and higher weights, can be studied in the context of matroid theory. They have shown that Betti numbers, of a matroid and its elongations, associated to a linear code, determine both the weight enumerator and the higher weights of the matroid. In the same vein, Britz, Johnsen, Mayhew and Shiromoto, introduced demimatroids as a natural generalization of matroids. They have shown that demimatroids are the appropriate combinatorial objects for studying Wei’s duality. Continuing in this line of research, we study the Hamming polynomial of a demimatroid, which is a generalization of the extended weight enumerator of a matroid. It results that the Hamming polynomial is a specialization of the Tutte polynomial of the demimatroid, and actually is equivalent to it. In addition, guided by the work of Johnsen, Roksvold and Verdure for matroids, we prove that Betti numbers of, a demimatroid and its elongations, determine the Hamming polynomial. Our results may be applied to simplicial complexes since in a canonical way they can be viewed as demimatroids. Furthermore, following work of Brylawski and Gordon, we show how demimatroids may be generalized one step further, to Brylawski structures. All this largely extends notions such as deletion, contraction, duality, nullity, minors, weight enumerators, higher weights, Tutte polynomial and MacWilliams identity, to non-matroidal structures.