Binary codes with covering radius one: Some new lower bounds (Q1377682)

This paper is devoted to the study of binary codes of length \(n\) with covering radius one via their characteristic functions. The author uses the method of congruence properties for sphere shells to get improvements on the lower bounds of \(K(n,1)\) (the minimum cardinality of such a code). The following special cases are treated: \(n\equiv 5\bmod 6\), \(n\equiv 1,3 \bmod 6\) and \(n\equiv 2,4\bmod 6\). The author gives an updated version of the lower bounds for \(K(n,1)\), \(n\leq 33\).