Counting the decimation classes of binary vectors with relatively prime length and density

We present a method for determining the number of decimation classes of density δ binary vectors indexed by a finite abelian group G of size ℓ and exponent ℓ ^* such that δ is relatively prime to ℓ ^* . This method is the first which is not based on exhaustive vector generation and exploits the subgroup lattice of ℤ_ℓ ^*^× . Instead, our method is based on our newly developed theory of multipliers for arbitrary subsets of finite abelian groups, our results on orbits under the action of the multiplier group, and finding the number of solutions of a potentially highly symmetric subset sum problem. Implementing our method on vectors indexed by ℤ_ℓ of odd length ℓ and density (ℓ +1)/2 greatly increased the number of ℓ for which the number of decimation classes of such vectors is known. Additionally, our newly developed theory provides information on the number of distinct translates fixed by each member of the multiplier group as well as sufficient conditions for each member of the multiplier group to be translate fixing.