Characterizing pyramidal Hadamard designs with the largest number of fixed points
Abstract
A symmetric (v,k,λ)-design is said to be f-pyramidal, with f<v−1, under the action of a group G if G acts as an automorphism group fixing f points and acting sharply transitively on the remaining ones. We show that, necessarily, f≤ v −2(k−λ). In particular, for a Hadamard design with parameters (v,k,λ)=(4m−1, 2m−1, m−1), for some m∈ℕ, it follows that f≤ 2m−1. Recently, working on the complement design, the family P of Hadamard $(4m-1,2m-1,m-1)$-designs admitting an abelian (2m−1)-pyramidal automorphism group G has been completely determined in the case m=2k>1. In this paper, we generalize that result by showing that P coincides with the family of Hadamard $(4m-1,2m-1,m-1)$-designs admitting a (2m−1)-pyramidal automorphism group, without assuming either that the group is abelian or that m is a power of $2$.
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BibTeX
@misc{traetta2026characterizing,
title = {{Characterizing pyramidal Hadamard designs with the largest number of fixed points}},
author = {Tommaso Traetta},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04858},
eprint = {2609.04858},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04858},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Characterizing pyramidal Hadamard designs with the largest number of fixed points
AU - Tommaso Traetta
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.04858
UR - https://arxiv.org/abs/2609.04858
AB - A symmetric (v,k,λ)-design is said to be f-pyramidal, with f<v−1, under the action of a group G if G acts as an automorphism group fixing f points and acting sharply transitively on the remaining ones. We show that, necessarily, f≤ v −2(k−λ). In particular, for a Hadamard design with parameters (v,k,λ)=(4m−1, 2m−1, m−1), for some m∈ℕ, it follows that f≤ 2m−1. Recently, working on the complement design, the family P of Hadamard $(4m-1,2m-1,m-1)$-designs admitting an abelian (2m−1)-pyramidal automorphism group G has been completely determined in the case m=2^k>1. In this paper, we generalize that result by showing that P coincides with the family of Hadamard $(4m-1,2m-1,m-1)$-designs admitting a (2m−1)-pyramidal automorphism group, without assuming either that the group is abelian or that m is a power of $2$.
ER -