On $p$-Divisibility of Weights in Codes and Cardinalities of Algebraic Sets
Dr. Daniel Katz
Princeton University

Abstract:

We consider past and recent progress on $p$-adic estimates of weights (i.e., estimates of weights modulo prime powers) in Abelian codes over finite fields, integer residue rings, and other Galois rings. Polynomial methods provide the framework for strengthening, generalizing, and unifying existing results, as well as proving the sharpness of some of the new results. We shall discuss connections to problems concerning algebraic sets over finite fields (specifically, the theorems of Chevalley, Warning, Ax, and N.M. Katz), and consider the problem of determining the size of algebraic sets for polynomials over integer residue rings.