Lower bounds for \(q\)-ary coverings by spheres of radius one (Q1331134)

Let \(C\) be a \(q\)-ary covering code with covering radius one. The author shows that if \((q - 1)n + 1\) does not divide \(q^ n\) and if \((q,n) \notin \{(2,2), (2,4)\}\), the sphere covering bound is not reached. The author characterizes the cases where the sphere covering bound is attained, when \(q\) is a prime power. The author also presents some improvements of lower bounds for binary and ternary codes.