Two techniques are presented for constructing new anticodes from known anticodes, namely the product and interleaving of anticodes. The product of anticodes (m1, k1, delta 1) and (m2, k2, delta 2) produces an (m1m2, k1k2, delta ) anticode, where delta 1 delta 21 delta 2,m2 delta 1). Interleaving of degree lambda of an (m, k, delta ) anticode produces an (m lambda , k lambda , delta lambda ) anticode. The efficiency of these constructions is examined in terms of the Griesmer bound for the binary case. As a result a rule is derived for selecting anticodes which can be efficiently combined either by product or interleaving.