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

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H. Liu , Chunming Tang , Cuiling Fan , Rong Luo ·

YOU? · · 2025 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2506.00429

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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Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/2506.00429
PDF: https://arxiv.org/pdf/2506.00429
OA Status: green
OpenAlex ID: https://openalex.org/W4414890345

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DOI: https://doi.org/10.48550/arxiv.2506.00429

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

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Type: preprint

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Language: en

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Publication year: 2025

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Publication date: 2025-05-31

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Authors: H. Liu, Chunming Tang, Cuiling Fan, Rong Luo

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Landing page: https://arxiv.org/abs/2506.00429

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PDF URL: https://arxiv.org/pdf/2506.00429

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Cited by: 0

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