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 van Wee. Let δ i denote the covering excess on a sphere of radius i , 0 ≤ i ≤ n . Generalizing an earlier result of van Wee, Habsieger and Honkala showed δ p − 1 ≥ p − 1 whenever n ≡ − 1 (mod p ) for an odd prime p and δ 0 = δ 1 = ⋯ = δ p − 2 = 0 holds. In the present paper we give the new estimation δ p − 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 conjectured by Habsieger.