An Upper Bound on the Volume of $q$-ary Codes

An analog of Sidel'nikov's lemma [Probl. Inf. Trans., 1974, vol. 10, no. 2, pp. 124–131] is obtained for $q$-ary codes with a specified vector composition. Upper bounds are obtained for the volume of codes with a specified vector composition and for arbitrary codes. An upper bound is obtained for the exponent of the volume of an arbitrary $q$-ary code that is uniformly superior to the Elias bound.