Counting the number of non-isotopic Taniguchi semifields

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.