Further results on planar DO functions and commutative semifields

It is proven that any Dembowski–Ostrom polynomial is planar if and only if its evaluation map is 2-to-1, which can be used to explain some known planar Dembowski–Ostrom polynomials. A direct connection between a planar Dembowski–Ostrom polynomial and a permutation polynomial is established if the corresponding semifield is of odd dimension over its nucleus. In addition, all commutative semifields of order 3⁵ are classified.     

Commutative presemifields and semifields

Strong conditions are derived for when two commutative presemifields are isotopic. It is then shown that any commutative presemifield of odd order can be described by a planar Dembowski-Ostrom polynomial and conversely, any planar Dembowski-Ostrom polynomial describes a commutative presemifield of odd order. These results allow a classification of all planar functions which describe presemifields isotopic to a finite field and of all planar functions which describe presemifields isotopic to Albert's commutative twisted fields. A classification of all planar Dembowski-Ostrom polynomials over any finite field of order p³, p an odd prime, is therefore obtained. The general theory developed in the article is then used to show the class of planar polynomials X¹⁰+aX⁶−a²X² with a≠0 describes precisely two new commutative presemifields of order e³ for each odd e?5.     

A geometric characterisation of Desarguesian spreads

We construct and describe the basic properties of a family of semifields in characteristic 2. The construction relies on the properties of projective polynomials over finite fields. We start by associating non-associative products to each such polynomial. The resulting presemifields form the degenerate case of our family. They are isotopic to the Knuth semifields which are quadratic over left and right nuclei. The non-degenerate members of our family display a very different behavior. Their left and right nuclei agree with the center, the middle nucleus is quadratic over the center. None of those semifields is isotopic or Knuth equivalent to a commutative semifield. As a by-product we obtain the complete taxonomy of the characteristic 2 semifields which are quadratic over the middle nucleus, bi-quadratic over the left and right nuclei and not isotopic to twisted fields. This includes determining when two such semifields are isotopic and the order of the autotopism group.     

The six semifield planes associated with a semifield flock

In 1965 Knuth (J. Algebra 2 (1965) 182) noticed that a finite semifield was determined by a 3-cube array (a_ijk) and that any permutation of the indices would give another semifield. In this article we explain the geometrical significance of these permutations. It is known that a pair of functions (f,g) where f and g are functions from GF(q) to GF(q) with the property that f and g are linear over some subfield and g(x)²+4xf(x) is a non-square for all x∈GF(q)∗, q odd, give rise to certain semifields, one of which is commutative of rank 2 over its middle nucleus, one of which arises from a semifield flock of the quadratic cone, and another that comes from a translation ovoid of Q(4,q). We show that there are in fact six non-isotopic semifields that can be constructed from such a pair of functions, which will give rise to six non-isomorphic semifield planes, unless (f,g) are of linear type or of Dickson-Kantor-Knuth type. These six semifields fall into two sets of three semifields related by Knuth operations.     

Albert's Construction for Semifields of Even Order

Albert's construction for commutative semifields of order 2 ⁿ , n odd, is presented. It avoids the construction of a presemifield and, in the case that n is prime, allows us to determine automorphism groups and the isomorphism classes. If n is a prime greater than three, the semifields are strictly not associative. These semifields are new for all n greater than three, differing from the binary semifields in that each admits only the trivial automorphism.

The authors present an explicit construction of an isotope of the 2⁵-element semifield that contains a subsemifield of order 2².     

Commutative semifields from projection mappings

We describe a general projection method to construct commutative semifields in odd characteristic. One application yields a family of commutative semifields of order q ^2m with middle nucleus of order at least q ² for every odd prime-power q and every odd integer m > 1. Another application of the method yields a generalization of the Budaghyan–Helleseth family and also greatly simplifies the construction.     

On a generalization of cyclic semifields

A new construction is given of cyclic semifields of orders q ²ⁿ , n odd, with kernel (left nucleus) and right and middle nuclei isomorphic to , and the isotopism classes are determined. Furthermore, this construction is generalized to produce potentially new semifields of the same general type that are not isotopic to cyclic semifields. In particular, a new semifield plane of order 4⁵ and new semifield planes of order 16⁵ are constructed by this method.     

Fractional dimensions in semifields of odd order

A finite semifield D is considered a fractional dimensional semifield if it contains a subsemifield E such that λ := log_|E||D| is not an integer. We develop spread-theoretic tools to determine when finite planes admit coordinatization by fractional semifields, and to find such semifields when they exist. We use our results to show that such semifields exist for prime powers 3 ⁿ whenever n is an odd integer divisible by 5 or 7.     

A new semifield of order 2

In [G. Lunardon, Semifields and linear sets of PG(1,q^t), Quad. Mat., Dept. Math., Seconda Univ. Napoli, Caserta (in press)], G. Lunardon has exhibited a construction method yielding a theoretical family of semifields of order q²ⁿ,n>1 and n odd, with left nucleus F_qⁿ, middle and right nuclei both F_q² and center F_q. When n=3 this method gives an alternative construction of a family of semifields described in [N.L. Johnson, G. Marino, O. Polverino, R. Trombetti, On a generalization of cyclic semifields, J. Algebraic Combin. 26 (2009), 1-34], which generalizes the family of cyclic semifields obtained by Jha and Johnson in [V. Jha, N.L. Johnson, Translation planes of large dimension admitting non-solvable groups, J. Geom. 45 (1992), 87-104]. For n>3, no example of a semifield belonging to this family is known.In this paper we first prove that, when n>3, any semifield belonging to the family introduced in the second work cited above is not isotopic to any semifield of the family constructed in the former. Then we construct, with the aid of a computer, a semifield of order 2¹⁰ belonging to the family introduced by Lunardon, which turns out to be non-isotopic to any other known semifield.     

A note on the Boerner-lantz semifield planes

We determine the number of nonisomorphic semifield planes of order p⁴ associated to the Boerner-Lantz semifields.     

Paley type group schemes and planar Dembowski–Ostrom polynomials

In this paper we give some necessary and sufficient conditions for Dembowski–Ostrom polynomials to be planar. These conditions give a simple explanation of the Coulter–Matthews and Ding–Yin commutative semifields and enable us to obtain permutation polynomials from some of the Zha–Kyureghyan–Wang commutative semifields. We then give a generalization of Feng’s construction of Paley type group schemes in extra-special p-groups of exponent p and construct a family of Paley type group schemes in what we call the flag groups of finite fields. We also determine the strong multiplier groups of these group schemes. In the last section of this paper, we give a straightforward generalization of the twin prime power construction of difference sets to a construction of Hadamard designs from twin Paley type association schemes.     

On isotopisms and strong isotopisms of commutative presemifields

In this paper we prove that the P(q,?) (q odd prime power and ?>1 odd) commutative semifields constructed by Bierbrauer (Des. Codes Cryptogr. 61:187?C196, 2011) are isotopic to some commutative presemifields constructed by Budaghyan and Helleseth (SETA, pp.?403?C414, 2008). Also, we show that they are strongly isotopic if and only if q??1(mod?4). Consequently, for each q???1(mod?4) there exist isotopic commutative presemifields of order q ^2? (?>1 odd) defining CCZ-inequivalent planar DO polynomials.     

On the Hughes–Kleinfeld and Knuth’s semifields two-dimensional over a weak nucleus

In 1960 Hughes and Kleinfeld (Am J Math 82:389–392, 1960) constructed a finite semifield which is two-dimensional over a weak nucleus, given an automorphism σ of a finite field and elements with the property that has no roots in . In 1965 Knuth (J Algebra 2:182–217, 1965) constructed a further three finite semifields which are also two-dimensional over a weak nucleus, given the same parameter set . Moreover, in the same article, Knuth describes operations that allow one to obtain up to six semifields from a given semifield. We show how these operations in fact relate these four finite semifields, for a fixed parameter set, and yield at most five non-isotopic semifields out of a possible 24. These five semifields form two sets of semifields, one of which consists of at most two non-isotopic semifields related by Knuth operations and the other of which consists of at most three non-isotopic semifields.      

On Strong Isotopy of Dickson Semifields and Geometric Implications

Similarity of certain abelian collineation groups of a translation plane corresponds to strong isotopy of multiplication variations of a commutative semifield. Strong isotopy of Dickson semifields and their multiplication variations is characterized. The splitting of the isotopy class of a Dickson semifield over different types of basic fields (e. g. absolutely algebraic fields, number fields) into classes of strong isotopy is investigated. **     

Semifield planes of order p 4 and kernel GF(p 2)

In this article we determine the number of non-isomorphic semifield planes of order p⁴ and kernel GF(p²) for p prime, 3 ≤ p ≤ 11. We show that for each of these values of p, the plane is either desarguesian, p-primitive, or a generalized twisted field plane. We also show that the class of p-primitive planes is the largest. We also discuss the autotopism group of the semifields under study.     

A class of semifields of order q4

This paper gives a class of semifields of order q⁴ for any prime power q = p^r with p greater than 3. It is shown that this class has left nucleus GF(q²), and right and middle nucleus GF(q). Although it is not proved, it is believed that all the semifields in this class are new.Supported in part by NSF     

Semifields and their properties

An introduction to the theory of semifields is included in the first part of the article: basic concepts, initial properties, and several methods of investigating semifields are examined. Semifields with a generator, in particular bounded semifields, are considered. Elements of the theory of kernels of semifields are also included in the paper: the structure of principal kernels; the kernel generated by the element 2 = 1 +1; indecomposable and maximal spectra of semifields; properties of the lattice of kernels of a semifield. A fragment of arp-semiring theory, which is the basis of a new method in semifield theory, is also included in the first part. The second part of the work is devoted to sheaves of semifields and functional representations of semifields. Properties of semifields of sections of semifield sheaves over a zero-dimensional compact are described. Two structural sheaves of semifields, which are the analogs of Pierce and Lambek sheaves for rings, are constructed. These sheaves give isomorphic functional representations of arbitrary, strongly Gelfand, and biregular semifields. As a result, sheaf characterizations of strongly Gelfand, biregular, and Boolean semifields are obtained.     

Classifying planar monomials over fields of order a prime cubed

Using Hermite's criteria, we classify planar monomials over fields of order a prime cubed, establishing the Dembowski-Ostrom conjecture for monomials over fields of such orders.