Counting the number of non-isotopic Taniguchi semifields

TL;DR

This paper provides a complete classification of Taniguchi semifields up to isotopy, establishing bounds on their quantity and demonstrating their status as the largest known family of semifields of odd order.

Contribution

It offers a complete characterization of isotopy classes of Taniguchi semifields and quantifies their total number, revealing their prominence among semifields of odd order.

Findings

Around p^{m+s} non-isotopic Taniguchi semifields of size p^{2m}

Taniguchi semifields form the largest known family of semifields of odd order

A new group-theoretic technique for determining isotopy using autotopism groups

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 size $p^{2m}$ where $s$ is the largest divisor of $m$ with $2s\neq m$. This result proves that the family of Taniguchi semifields is (asymptotically) the biggest 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.