Almost difference sets and reversible divisible difference sets
Almost difference sets are divisible difference sets with the property that \(\lambda_ 1\) and \(\lambda_ 2\) differ by 1 (they would be difference sets if \(\lambda_ 1=\lambda_ 2\)). Two constructions are given. The first set of parameters are \((4,3^{2a},2(3^{2a}-3^ a),3^{2a}- 2\cdot 3^ a,3^{2a}-2\cdot 3^ a+ 1)\), and these are similar to \((4\cdot 3^{2a},2\cdot 3^{2a}- 3^ a,3^{2a}-3^ a)\) (Menon) difference sets. The second construction is built by using the hyperplanes of EA\((q^{d+1})\) together with a \((m,n,h,h-1,\lambda_ 2)\) divisible difference set to get a \((m,nq^{d+1},q^ d h,q^ d\Bigl({q^ d-1\over q-1}\Bigr),q^{d-1}\lambda_ 2) \text{DDS}\). This is not an almost difference set in general, but we note two special cases where it is an almost difference set. We give two modifications of this basic construction, and we also give examples where this construction is reversible.
scientific article; zbMATH DE number 572365