Lower Bounds for q-ary Codes with Large Covering Radius

Let Kq(n,R) denote the minimal cardinality of a q-ary code of length n and covering radius R. Recently the authors gave a new proof of a classical lower bound of Rodemich on Kq(n,n−2) by the use of partition matrices and their transversals. In this paper we show that, in contrast to Rodemich’s original proof, the method generalizes to lower-bound Kq(n,n − k) for any k> 2. The approach is bestunderstood in terms of a game where a winning strategy for one of the players implies the non-existence of a code. This proves to be by far the most efficient method presently known to lower-bound Kq(n,R) for large R (i.e. small k). One instance: the trivial sphere-covering bound K12(7,3) � 729, the previously best bound K12(7,3) � 732 and the new bound K12(7,3) � 878