Subgeometry partitions from cyclic semifields

Abstract

New cyclic semifield planes of order q^lcm(m,n) are constructed. By varying m and n, while preserving the lcm(m,n), necessarily mutually non-isomorphic semifield planes are obtained. If lcm(m,n)/m = 3, new GL(2,q^m) - q^3m-planes are constructed. If m is even, new subgeometry partitions in PG(lcm(m, n)-1, q²), by subgeometries isomorphic to either PG(lcm(m,n)/2-1, q²) or PG(lcm(m,n)-1, q) are constructed. If the 2-order of m is strictly larger than the 2-order of n then ‘double’ retraction is possible producing two distinct subgeometry partitions from the same semifield plane. If m is even and lcm(m,n)/m = 3, new subgeometry partitions may be constructed from the GL(2,q^m) - q^3m-planes.