BCH codes and distance multi- or fractional colorings in hypercubes asymptotically

Abstract

The main result is a short and elementary proof for the author's exact asymptotic results on distance chromatic parameters (both number and index) in hypercubes. Moreover, the results are extended to those on fractional distance chromatic parameters and on distance multi-colorings. Inspiration comes from radio frequencies allocation problem. The basic idea is the observation that binary primitive narrow-sense BCH codes or their shortenings have size asymptotically within a constant factor below the largest possible size, A ( n , d ) , among all binary codes of the same length, n, and the same minimum distance, d, as n → ∞ while d is constant. Also a lower bound in terms of A ( n , d ) is obtained for B ( n , d ) , the largest size among linear binary codes.