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arXiv · 2506.00429

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

Hengfeng Liu·Chunming Tang·Cuiling Fan·Rong Luo
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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.

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Hengfeng Liu, Chunming Tang, Cuiling Fan, Rong Luo. 2025-05-31. Combinatorial $t$-Designs from Finite Abelian Groups and Their Applications to Elliptic Curve Codes. https://arxiv.org/abs/2506.00429

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