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

TL;DR

This paper characterizes conditions under which finite abelian groups and their subset families form $t$-designs, and applies these results to construct elliptic curve codes with design properties from subset sums.

Contribution

It provides new necessary and sufficient conditions for $t$-designs in finite abelian groups and links these designs to elliptic curve codes via subset sums.

Findings

Characterization of $1$-designs in finite abelian $p$-groups.

Conditions for $1$-designs in groups of exponent $pq$.

Establishment of a correspondence between $t$-designs and elliptic curve codes.

Abstract

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.