On Perfect 3{ and 4{Shift N{Designs

In 1] Levenshtein and Vinck investigated perfect run{length{limited codes which are capable of correcting single peak shifts. As a basic combinatorial tool for the construction of such codes they introduced the concept of a t{shift N{design, which is deened to be an N{elementary subset H = fh 1 ; : : : h N g of a nite additive Abelian group G, with the property that for any = 1; : : : t and any i = 1; : : : ; N all 2Nt elements h i are diierent and not equal to zero. Such a design is said to be perfect if jGj = 2tN + 1, i. e., for every element g 2 G there are exactly one h i 2 H and 2 f1; : : : ; tg such that g = h i or g = ? h i. Vinck and Levenshtein 1] also gave explicit constructions of perfect t{shift designs for the special values t = 1; 2 and for the case t = p?1 2 , N = (p ? 1)=(p ? 1), where p is a prime number. We shall now investigate perfect t{shift N{designs in Z 2tN+1 , the ring of integers modulo 2tN + 1, for the parameters t = 3 and t = 4. In order to do so, we shall rst introduce a more general shift design concept.