Academic Journal
Primitive rank 3 groups, binary codes, and 3-designs
| Title: | Primitive rank 3 groups, binary codes, and 3-designs |
|---|---|
| Authors: | Rodrigues, B, Solé, Patrick |
| Contributors: | Solé, Patrick |
| Source: | Designs, Codes and Cryptography. 93:3463-3479 |
| Publisher Information: | Springer Science and Business Media LLC, 2025. |
| Publication Year: | 2025 |
| Subject Terms: | [MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO], rank 3 groups, BIBD, PBIBD, 94B15, [INFO.INFO-IT] Computer Science [cs]/Information Theory [cs.IT], 05E30 |
| Description: | Let G be a primitive rank 3 permutation group acting on a set of size v. Binary codes of length v globally invariant under G are well-known to hold PBIBDs in their $$A_w$$ A w codewords of weight w. The parameters of these designs are $$\bigg (A_w,v,w,\frac{wA_w}{v},\lambda _1,\lambda _2\bigg ).$$ ( A w , v , w , w A w v , λ 1 , λ 2 ) . When $$\lambda _1=\lambda _2=\lambda ,$$ λ 1 = λ 2 = λ , the PBIBD becomes a 2- $$(v,w,\lambda )$$ ( v , w , λ ) design. We obtain computationally 111 such designs when G ranges over $$\textrm{L}_2(8){:}3, \textrm{U}_{4}(2), \textrm{U}_{3}(3){:}2, \textrm{A}_8, \textrm{S}_6(2),$$ L 2 ( 8 ) : 3 , U 4 ( 2 ) , U 3 ( 3 ) : 2 , A 8 , S 6 ( 2 ) , $$\textrm{S}_{4}(4), \textrm{U}_{5}(2), \textrm{M}_{11}, \textrm{M}_{22}, \textrm{HS}, \textrm{G}_2(4), \textrm{S}_{8}(2),\textrm{O}^{+}_{10}(2),$$ S 4 ( 4 ) , U 5 ( 2 ) , M 11 , M 22 , HS , G 2 ( 4 ) , S 8 ( 2 ) , O 10 + ( 2 ) , and $$\textrm{O}^{-}_{10}(2)$$ O 10 - ( 2 ) in the notation of the Atlas. Included in the counting are 2-designs which are held by nonzero weight codewords of the binary adjacency codes of the triangular and square lattice graphs, respectively. The 2-designs in this paper can be obtained neither from Assmus–Mattson theorem, nor by the classical 2-tra nsitivity (or 2-homogeneity) argument of the automorphism group of the code. Further, the extensions of the codes that hold 2-designs sometimes hold 3-designs. We thus obtain nine self-complementary 3-designs on 16 (4), $$28,\, 36$$ 28 , 36 (2), $$\,56,\, 176$$ 56 , 176 points respectively. The design on 176 points is invariant under the Higman–Sims group. |
| Document Type: | Article |
| File Description: | application/pdf |
| Language: | English |
| ISSN: | 1573-7586 0925-1022 |
| DOI: | 10.1007/s10623-025-01647-3 |
| Access URL: | https://hal.science/hal-05054574v1 https://hal.science/hal-05054574v1/document https://doi.org/10.1007/s10623-025-01647-3 |
| Rights: | CC BY |
| Accession Number: | edsair.doi.dedup.....1f20fd6d1e9f2a08e61e63bbdd3e60d5 |
| Database: | OpenAIRE |
| ISSN: | 15737586 09251022 |
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| DOI: | 10.1007/s10623-025-01647-3 |