Isotopy classification of the new semifield family when parameters coincide with known constructions

Determine whether the semifields arising from the family D_{n,s,1}(\gamma,F) (denoted in the paper as the family with parameters (q^{2ts}, q^{t}, q^{t}, q^{s}, q), constructed via the multiplication a \star_{D} b = (a - (\gamma/f_0) a_0'' f) b in the quotient R/Rf over the cyclic Galois extension \mathbb{F}_{q^n}/\mathbb{F}_q with n=2t and an irreducible central polynomial F(y) of degree s) are isotopic to previously known semifields when their nuclear parameters match; in particular, ascertain for which choices of n, t, s, and \gamma these semifields are genuinely new despite sharing parameters with existing constructions.

Background

The paper introduces a second new family of MRD codes and semifields D_{n,s\ell,k}(\gamma,F), extending the Hughes–Kleinfeld/Trombetti–Zhou constructions to both finite and infinite fields. In the finite-field case with n=2t and s=deg(F), setting k=1 yields a family of semifields of order q^{2ts} whose right nucleus is \mathbb{F}{q^s} and whose left and middle nuclei are \mathbb{F}{q^t}.

The authors prove newness for an infinite range of parameters by comparing invariants (idealisers, centralisers, and centres) but note that when the nuclear parameters coincide with those of known families, deciding isotopy is difficult. They explicitly leave the question of whether these semifields are new in parameter-overlap cases as an open problem.