Combinatorial t-Designs from Finite Abelian Groups and Their Applications to Elliptic Curve Codes

In this paper, we establish the conditions for some finite abelian groups and the family all the k-sets in each of them summing up to an element x to form t-designs. We fully characterize the sufficient and necessary conditions for the incidence structures to form $1$-designs in finite abelian p-groups, generalizing existing results on vector spaces over finite fields. For finite abelian groups of exponent $pq$, we also propose sufficient and necessary conditions for the incidence structures to form a $1$-designs. Furthermore, some interesting observations of the general case when the group is cyclic or non-cyclic are presented and the relations between $(t-1)$-designs and t-designs from subset sums are established. As an application, we demonstrate the correspondence between t-designs from the minimum-weight codewords in elliptic curve codes and subset-sum designs in their groups of rational points. By such a correspondence, elliptic curve codes supporting designs can be simply derived from subset sums in finite abelian groups that supporting designs.