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On Finite Semifields with a Weak Nucleus and a Designed

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On Finite Semifields with a Weak Nucleus and a Designed On finite semifields with a weak nucleus and a designed automorphism group A. Pi~nera-Nicol´as∗ I.F. R´uay Abstract Finite nonassociative division algebras S (i.e., finite semifields) with a weak nucleus N ⊆ S as defined by D.E. Knuth in [10] (i.e., satisfying the conditions (ab)c − a(bc) = (ac)b − a(cb) = c(ab) − (ca)b = 0, for all a; b 2 N; c 2 S) and a prefixed automorphism group are considered. In particular, a construction of semifields of order 64 with weak nucleus F4 and a cyclic automorphism group C5 is introduced. This construction is extended to obtain sporadic finite semifields of orders 256 and 512. Keywords: Finite Semifield; Projective planes; Automorphism Group AMS classification: 12K10, 51E35 1 Introduction A finite semifield S is a finite nonassociative division algebra. These rings play an important role in the context of finite geometries since they coordinatize projective semifield planes [7]. They also have applications on coding theory [4, 8, 6], combinatorics and graph theory [13]. During the last few years, computational efforts in order to clasify some of these objects have been made. For instance, the classification of semifields with 64 elements is completely known, [16], as well as those with 243 elements, [17]. However, many questions are open yet: there is not a classification of semifields with 128 or 256 elements, despite of the powerful computers available nowadays. The knowledge of the structure of these concrete semifields can inspire new general constructions, as suggested in [9]. In this sense, the work of Lavrauw and Sheekey [11] gives examples of constructions of semifields which are neither twisted fields nor two-dimensional over a nucleus starting from the classification of semifields with 64 elements. In particular, these are the semifields labelled as XIV, XXIV, XXVI and XXVII in [16]. On the other hand, the construction of semifields with a prescribed automorphism group has been carried out in references such as [3] and [14]. The present work is motivated by the behaviour of some semifields with 64 elements which also appear in the classification of [16]. In particular, semifields coordinatizing 18 planes in the Knuth classes XVII and XVIII and 27 in the Knuth class XXXIX (see Table 5 of [16]) have an automorphism group isomorphic to the group C5. Notice that the order of this group, 5, and the dimension of the semifields over their center, 6, are relatively prime (a fact that differs completely from the situation of the finite field F64). We will see in the next section that all these semifields can be additively described as a direct sum of two Galois fields of orders 4 and 16. Moreover, some of them can be seen as a 3−dimensional vector spaces over a so called weak nucleus, see [10]. ∗Institute of Mathematics (IMUVa) and Applied Mathematics Department, Universidad de Valladolid, [email protected] yDepartamento de Matem´aticas,Universidad de Oviedo, [email protected] 1 In the last part of the paper, we will use this concept in order to generalize the construction of such semifields. Exploring these ideas, we will be able to construct sporadic examples of semifields with 256 and 512 elements whose automorphism groups contain a subgroup of prescribed structure. These semifields are new, since they do not fall in the Knuth orbit of any finite semifield either listed in [12] or contructed by [18]. 2 Semifields of order 64 with a cyclic automorphism group C5 In this section we will study the structure of a semifield S with 64 elements with cyclic automorphism group C5. First of all, observe that the center of any such a finite semifield contains F2 = f0; 1g, and S is naturally a 6-dimensional F2−algebra. We will prove that it is possible to find a set of 4 elements fixed by the action of the automorphism group, which yields an alternative additive decomposition of S. Theorem 1. Let (S; +; ∗) be a semifield with 64 elements and automorphism group Aut(S) = h'i isomorphic to C5. Then, there exists an element a 6= 0; 1, such that (f0; 1; a; a + 1g; +; ∗) is a semifield isomorphic to F4 which is fixed by the automorphism group. Moreover, there exists an element b 2 i i j i+j S n f0; 1; a; a + 1g such that (f' (b) j i 2 Ng; +; ·), with ' (b) · ' (b) = ' (b), is isomorphic to the finite field F16. So, S can be seen as a F2-vector space isomorphic to the direct sum F4 ⊕ F16, with basis f1; a; b; '(b);'2(b);'3(b)g. Proof. Let us consider the set Fix(') = fx 2 S j '(x) = xg of fixed points under the action of Aut(S) over S. Since jAut(S)j = 5, we have 64 6= jFix(')j ≡ 64 (mod 5). So, jFix(')j = 4 and, because ∼ there is a unique semifield with four elements, (Fix('); +; ∗) = F4. Therefore, we can find an element a 2 S n f0; 1g such that Fix(') = f0; 1; a; a + 1g. 5 5 4 3 2 Since ' 6= I = ' and x + 1 = (x + 1)(x + x + x + x + 1), then the F2-irreducible polynomial x4 +x3 +x2 +x+1 divides the minimal polynomial of '. So, there exist an element b 2 S nf0; 1; a; a+ 2 3 1g such that the F2[x]−cyclic submodule generated by b, hb; '(b);' (b);' (b)iF2 , is isomorphic to 4 3 2 ∼ F2[x]=hx + x + x + x + 1i = F16. 2 3 Finally, notice that dimF2 S = 6, so the set f1; a; b; '(b);' (b);' (b)g, which is linearly independent over F2, is an F2−basis of S. Remark 1. The decomposition of S as F4 ⊕ F16 corresponds to the F2−module decomposition 2 4 3 2 (F2[x]=hx + 1i) ⊕ F2[x]=hx + x + x + x + 1i and that ' preserves this decomposition. Notice that the restriction of the product ∗ of S to Fix(') is the product of the field F4, but this does not happen with F16. However, ξ = '(b) 2 F16 can be identified with a 5th−primitive root of unity, and it can be assumed that the action of the automorphism ' over the elements of F16 is given by '(c) = ξ · c for all c 2 F16. From now on we shall simply write ξc instead of ξ · c. The next step in the description of semifields with 64 elements with cyclic automorphism group C5 consists on the introduction of the definition of weak nucleus in the sense of [10]. Definition 1. Let (S; +; ∗) be a semifield and let F ⊆ S be a field. Then, F is a weak nucleus for S if (a ∗ b) ∗ c = a ∗ (b ∗ c) whenever any two of a; b; c are in F. Notice that if there exists a weak nucleus F of S, then S can be written as an F-vector space. However, right and left multiplications (Ra(x) = x ∗ a; La(x) = a ∗ x; for all a; x 2 S) will not be, in general, F-linear transformations. Moreover, the weak nucleus is generally not preserved by isotopy. 2 In order to describe the structure of the semifields with weak nucleus presented in this section, we need the following result, which will be stated without proof. Lemma 1. [Theorem 4.1 of [15]] Let R = GR(qd; pd) be a Galois ring of order qd and characteristic d p (in particular a finite field). For any finite (R; R)-bimodule RMR there exist a generator system fµ1; : : : ; µkg (called distinguished basis) and a system of automorphism σ1; : : : ; σk 2 Aut(R) such that 8a 2 R; l 2 f1; : : : ; kg : µla = σl(a)µl and M = Rµ1 ⊕ · · · ⊕ µk is a direct sum of cyclic (R; R)-bimodules. ∼ Theorem 2. Let (S; +; ∗) be a semifield with 64 elements, automorphism group Aut(S) = h'i = C5 ∼ and weak nucleus F = Fix(') = F4. Then, there exists an element λ 2 S such that f1; λ, '(λ)g is an F4-basis of S. Moreover, either a ∗ λ = λ ∗ a for all a 2 F, and the basis is called of type A, or 2 a ∗ λ = λ ∗ a for all a 2 F, and it is called of type B. Proof. From Definition 1, it follows that the semifield S is a bimodule over the weak nucleus F. So, by Lemma 1, there exists a distinguished F-basis of S, that is, it is possible to find an F-basis fs1; s2; s3g of S such that, for 1 ≤ i ≤ 3, a ∗ si = si ∗ σi(a), with σi 2 Aut(F), for all a 2 F. Notice that we can always take s1 = 1 and σ1 = Id. If si = 1 for some i, the claim is trivial. Otherwise, it suffices to see that f1; s2; s3g is a generator system from S whith a ∗ 1 = 1 ∗ a for all a 2 F. Let us denote the element s2 by λ. Suppose that σ2 = Id and so, a ∗ λ = λ ∗ a for all a 2 F. Since S is isomorphic, as F2−vector space, to the direct sum F ⊕ F16, we can assume that λ 2 F16. Otherwise, λ = A + B, with A 2 F and B 2 F16, and f1; λ − A; s3g is also a distinguished F-basis of S. Under these assumptions, we claim that f1; λ, '(λ)g is an F−distinguished basis of S. Since λ 2 F16, 4 3 2 P4 k the minimal polynomial '−annihilating λ is x + x + x + x + 1, i.e, k=0 ' (λ) = 0.
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