Abstract
We investigate the isotopy question for Taniguchi semifields. We give a complete characterization when two Taniguchi semifields are isotopic. We further give precise upper and lower bounds for the total number of non-isotopic Taniguchi semifields, proving that there are around \(p^{m+s}\) non-isotopic Taniguchi semifields of order \(p^{2m}\) where s is the largest divisor of m with \(2\,s\ne m\). This result proves that the family of Taniguchi semifields is (asymptotically) the largest known family of semifields of odd order. The key ingredient of the proofs is a technique to determine isotopy that uses group theory to exploit the existence of certain large subgroups of the autotopism group of a semifield.
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Acknowledgements
The athors would like to thank William Kantor and Yue Zhou for their comments, and bringing some literature on non-commutative semifields referred to in Conclusion to their attention. We further thank an anonymous reviewer for spotting a calculation error in Proposition 3. The first author is supported by GAČR Grant 18-19087 S - 301-13/201843. The second author is supported by NSF Grant 2127742.
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Göloğlu, F., Kölsch, L. Counting the number of non-isotopic Taniguchi semifields. Des. Codes Cryptogr. 92, 681–694 (2024). https://doi.org/10.1007/s10623-023-01262-0
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DOI: https://doi.org/10.1007/s10623-023-01262-0