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

Abstract

Abstract 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 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. The author presents a new technique for proving the 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 partially already conjectured by Habsieger.