Enumerative combinatorics and coding theory (Q1332388)

The author develops a new method of investigation of combinatorial problems, introducing the value enumerator \(V_ f(T)= \sum_ p T^{f(p)}\in \mathbb{N}[T,T^{-1}]\) \((p\in \{1,-1\}^ n)\) for a certain polynomial \(f\) in \(n\) variables with non-negative integral coefficients. The coefficient of \(T^ v\) is the number of binary points \(p\) such that \(f(p)= v\) \((v\in \mathbb{Z})\); an important partial case is \(v= 0\) (binary zeros). Finding \(V_ f(T)\) is shown to be equivalent to the enumeration of weights in some associated binary linear code. This correspondence, together with the MacWilliams identity for the weight distribution, is used to enumerate Hadamard matrices of some fixed order (the first known result of this kind), as well as the proper 4-colorings of a graph.