In this paper we show first that finding the location of a test vector in the sequence generated by an accumulator driven with an odd additive constant C is equivalent to the solution of a linear Diophantine equation with two variables. The latter equation is known to be solved fast in linear time. We then show that only one Diophantine equation needs to be solved per test set irrespective of the number of patterns in it. The finding of the locations of all patterns of a given test set T in the sequence generated under a constant C is done in O(n+|T|) steps instead of O(n ? |T|) steps of a previous approach. Next we present a method which, given a test set T, and an odd constant C, finds the seed in the sequence generated under C that can reproduce all patterns of T in minimum length. We use this optimum technique to search for the best constant C? (in terms of short test length) in a randomly generated subset. Experimental results show the potential of the approach for test set embedding.