Classification of additive-map pairs underpinning generalized MRD/semifield constructions

Classify all pairs of additive maps \phi_1, \phi_2: L \to L over a cyclic Galois extension L/K of degree n (with skew polynomial ring R=L[x;\sigma] and irreducible central polynomial F(y) of degree s, where F_0 is the constant coefficient of F(x^n)) that satisfy N_{L/K}(\phi_1(a)) \neq N_{L/K}(\phi_2(a))\,(-1)^{sk\ell(n-1)}\,F_0^{k\ell} for all a \in L, particularly in the case where L is infinite, so as to systematize the resulting families of semifields and MRD codes obtained from such pairs.

Background

Beyond the explicit families constructed in the paper, the authors note that one can generate further semifields and MRD codes from any pair of additive maps \phi_1, \phi_2 obeying a specific norm-inequality constraint tied to the central polynomial F(x^n) and parameters s, k, and \ell=n/m.

While this framework potentially yields broad classes of new structures, organizing and classifying all such admissible pairs—especially over infinite fields—remains unresolved, and the authors explicitly flag this as an open problem.