Partial sums of binomials, intersecting numbers, and the excess bound in Rosenbloom–Tsfasman space

Abstract

In this work, the sphere-covering bound on covering codes in Rosenbloom–Tsfasman spaces (RT spaces) is improved by generalizing the excess counting method. The approach focuses on studying the parity of a Rosenbloom–Tsfasman sphere (RT sphere) and the parity of the intersection of two RT spheres. We connect the parity of an RT sphere with partial sums of binomial coefficients and p-adic valuation of binomial coefficients. The intersection number of RT spaces is introduced and we determinate its parity under some conditions. Numerical applications of the method are discussed.