The technique of error-trapping decoding for algebraic codes is studied in combinatorial terms of covering systems. Letn, k, andtbe positive integers such thatn geq k geq t > 0. An(n, k,t)-covering system is a pair(X, beta), whereXis a set of sizenandbetais a collection of subsets ofX, each of sizek, such that for allT subseteq Xof sizet, there exists at least oneB in betawithTsubseteq B. Letb(n, k, t)denote the smallest size ofbeta, such that(X, beta)is an(n, k, t)-covering system. It is shown that the complexity of an error-trapping decoding technique is bounded byb(n, k, t)from below. Two new methods for constructing small(n, k, t)-covering systems, the algorithmic method and the difference family method, are given.