Publication at Faculty of Mathematics and Physics |
2023
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 ? 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.