Combinatorial $t$-Designs from Finite Abelian Groups and Their Applications to Elliptic Curve Codes
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| Format: | Preprint |
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2025
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| author | Liu, Hengfeng Tang, Chunming Fan, Cuiling Luo, Rong |
| author_facet | Liu, Hengfeng Tang, Chunming Fan, Cuiling Luo, Rong |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_00429 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Combinatorial $t$-Designs from Finite Abelian Groups and Their Applications to Elliptic Curve Codes Liu, Hengfeng Tang, Chunming Fan, Cuiling Luo, Rong Combinatorics Information Theory 05B05, 94B05 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. |
| title | Combinatorial $t$-Designs from Finite Abelian Groups and Their Applications to Elliptic Curve Codes |
| topic | Combinatorics Information Theory 05B05, 94B05 |
| url | https://arxiv.org/abs/2506.00429 |