Commutative semifields of order 243 and 3125

Abstract

This note summarises a recent search for commutative semifields of order 243 and 3125. For each of these two orders, we use the correspondence between commutative semifields of odd order and planar Dembowski-Ostrom polynomials to classify those commutative semifields which can be represented by a planar DO polynomial with coefficients in the base field. The classification yields a new commutative semifield of each order. Furthermore, the new commutative semifield of order 243 describes a skew Hadamard difference set which is also new.

Symplectic spreads, planar functions, and mutually unbiased bases

“…where F = F q , q = 3 t , and t ≥ 3 odd . The Penttila-Williams sporadic symplectic semifield [40] (F 3 5 ⊕ F 3 5 , +, •) of order 3 10 and the corresponding commutative semifield are given by During last several years some families of quadratic planar functions were discovered [10,12,15,16,17,20,47,49,50,52].…”

“…During last several years some families of quadratic planar functions were discovered [10,12,15,16,17,20,47,49,50,52].…”

Symplectic 4-dimensional semifields of order $$8^4$$ and $$9^4$$

“…A computational classification of all semifields of order 32 was obtained in 1962 (using a CDC 1604) by Walker [25], and confirmed by Knuth [10] soon after. More recently, computational classifications of semifields were obtained by Dempwolff [6] (q n = 3 4 ), and by Combarro, Ranilla, and Rúa, see [20] (q n = 2 6 ), [21] (q n = 4 4 , 5 4 ), [22] (q n = 3 5 ), and [23] (q n = 7 4 ).…”

“…Complete computational classifications are also known for certain special type of semifields, see for example [14], in which all 8-dimensional rank 2 commutative semifields are classified, relying on theoretical results from [3] and [12]. A partial classification of commutative semifields of order 3 5 and 5 5 can be found in [5], where the authors restrict the classification to commutative semifields whose corresponding Dembowski-Ostrom polynomials have their coefficients in the base field. Theoretical results on rank 3 commutative semifields of dimension at least 6 can be found in [4,16,19].…”