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A Family of Gauss Type Hadamard Difference Sets
Abstract
A Hadamard difference set (HDS) D of order u2 in an abelian group G satisfies |χ(D)|=u for every nontrivial character χ of G. We call such a character value naive if it is divisible by u, i.e., if it is equal to u times a root of unity. All previously known abelian HDSs only have naive character values. We show that for d≥ 1 and u=3d, a group Z32× H, with H an abelian group of order 22d+2, contains a HDS of order u2 with non-naive character values if and only if 8≤exp H≤ 2d+2. All difference sets obtained are new. The proof rests on a specific HDS in Z32× Z8× Z2, a covering extended building set on Z32× Z8× Z4, and a variation of the Davis-Jedwab recursive construction.
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BibTeX
@misc{schmidt2026family,
title = {{A Family of Gauss Type Hadamard Difference Sets}},
author = {Bernhard Schmidt},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07088},
eprint = {2609.07088},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07088},
}
RIS
TY - UNPB
TI - A Family of Gauss Type Hadamard Difference Sets
AU - Bernhard Schmidt
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.07088
UR - https://arxiv.org/abs/2609.07088
AB - A Hadamard difference set (HDS) D of order u^2 in an abelian group G satisfies |χ(D)|=u for every nontrivial character χ of G. We call such a character value naive if it is divisible by u, i.e., if it is equal to u times a root of unity. All previously known abelian HDSs only have naive character values. We show that for d≥ 1 and u=3d, a group Z_3^2× H, with H an abelian group of order 2^2d+2, contains a HDS of order u^2 with non-naive character values if and only if 8≤exp H≤ 2^d+2. All difference sets obtained are new. The proof rests on a specific HDS in Z_3^2× Z_8× Z_2, a covering extended building set on Z_3^2× Z_8× Z_4, and a variation of the Davis-Jedwab recursive construction.
ER -