On the general excess bound for binary codes with covering radius one

Let K(n,1) denote the minimal cardinality of a binary code of length n and covering radius one. Fundamental for the theory of lower bounds for K(n,1) is the covering excess method introduced by Johnson and v. Wee. Let δi denote the covering excess on a sphere of radius i, 0⩽i⩽n. Generalizing an earlier result of v. Wee Habsieger and Honkala showed δp−1⩾p−1 whenever n≡−1(modp) for an odd prime p and δ0=δ1=...=δp−2=0 holds.The author presents a new technique for proving the estimation δp−1⩾(p−2)p−1 instead. This answers a question of Habsieger and yields a “general improvement of the general excess bound” for binary codes with covering radius one. The proof uses a classification theorem for certain subset systems as well as new congruence properties for the δ-function, which were partially already conjectured by Habsieger.