New lower bounds for covering codes

We develop two methods for obtaining new lower bounds for the cardinality of covering codes. Both are based on the notion of linear inequality of a code. Indeed, every linear inequality of a code (defined on F-q(n)) allows to obtain, using a classical formula (inequality (2) below), a lower bound on K-q(n, R), the minimum cardinality of a covering code with radius R. We first show how to get new linear inequalities (providing new lower bounds) from old ones. Then, we prove some formulae that improve on the classical formula (2) for linear inequalities of some given types. Applying both methods to all the classical cases of the literature, we improve on nearly 20% of the best lower bounds on K-q(n, R). (C) 2000 Elsevier Science B.V. All rights reserved.