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

Let K(n,1)K(n,1) denote the minimal cardinality of a binary code of length nn and covering radius one. Fundamental for the theory of lower bounds for K(n,1)K(n,1) is the covering excess method introduced by Johnson and van Wee. Let δiδi denote the covering excess on a sphere of radius ii, 0≤i≤n0≤i≤n. Generalizing an earlier result of van Wee, Habsieger and Honkala showed δp−1≥p−1δp−1≥p−1 whenever n≡−1n≡−1 (mod pp) for an odd prime pp and δ0=δ1=⋯=δp−2=0δ0=δ1=⋯=δp−2=0 holds. In the present paper we give the new estimation δp−1≥(p−2)p−1δp−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 δδ-function, which were conjectured by Habsieger.