Ovals in $${\mathbb {Z}}^2_{2p}$$ Z 2 p 2
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Annals of Combinatorics
Ovals in Z22p Zofia St¸epie´ n Abstract. By an oval in Z22p , p odd prime, we mean a set of 2p + 2 points, such that no three of them are on a line. It is shown that ovals in Z22p only exist for p = 3, 5 and they are unique up to an isomorphism. Mathematics Subject Classification. 05B99. Keywords. Arc, Collinearity.
1. Introduction A k-arc is a set of k points, such that no three of them are collinear. Arcs are well studied in projective geometry, see, e.g., [1,9] and [2] for further references. Recall that any non-singular conic of PG(2, q) is a (q + 1)-arc. If K is any karc of PG(2, q) with q odd, then k ≤ q + 1. A (q + 1)-arc is called an oval. A famous theorem of Segre [9] tells us that for q odd, every oval of PG(2, p) is a non-singular conic. Many authors continue to study the classical problem in the context of Hjelmslev geometry, see, e.g., [3,4,8] and [6] for the definition of the abstract Hjelmslev plane. In this article, we consider similar questions in Z2n . This modified problem is still interesting and it was investigated in [5,7] and [10]. If n = p is prime, the resulting space is just AG(2, p). The correspondence between projective and affine planes leads to the following facts: The maximum size of an arc in Z2p is p + 1; every (p + 1)-arc in Z2p is a non-singular conic. The maximum size of a cap in Z22p , p odd prime, is 2p + 2 (see Lemma 4.1 in [7] and Theorem 3.1 in [10]). We will call a (2p + 2)-arc in Z22p an oval in Z22p . In this article, we completely solve the problem of the existence and the uniqueness of ovals in Z22p . We define a line in Z2n to be a subset of Z2n of the form: {(x; y) : ax + by + c = 0} , where gcd(a; b; n) = 1. This line is denoted by [a;b;c] . 0123456789().: V,-vol
Z. St¸ epie´ n
By an automorphism of Z2n , we mean a mapping from Z2n to Z2n which preserves arcs. Let X, Y be arcs in Z2n . We say that X and Y are isomorphic if there exists an automorphism of Z2n mapping X to Y.
2. The General Case Due to the Chinese remainder theorem, we have the following. Lemma 2.1. Let p be an odd prime, φ2 : Z22p → Z22 , φp : Z22p → Z2p be reduction maps. Then, any three points x, y, z ∈ Z22p are collinear if and only if φ2 (x), φ2 (y), φ2 (z) ∈ Z22 and φp (x), φp (y), φp (z) ∈ Z2p are collinear. From now on, we will use the notation of the previous Lemma. Lemma 2.2. Let p be an odd prime. Let σ be an arbitrary permutation of Z22 . Then, there exists an automorphism F of Z22p , such that φ2 ◦ F = σ ◦ φ2 and φp ◦ F = Id ◦ φp . FB be linear transformations determined by matrices A = Proof. p+1 1 0 Let FA and p and B = p1 p p+1 , respectively. Denote by t[0,p] the translation by a vector [0, p]. Note that the group of permutations of the set Z22 is generated by transpositions σ1 = ((0,0),(0,1)) , σ2 = ((0,1),(1,0)) , σ3 = ((1,0),(1,1)) . One can verify by a straightforward calculation that φ2 ◦ FA ◦ t[0,p] = σ1 ◦ φ2 , φ2 ◦ FB = σ2 ◦ φ2 and φ2 ◦ FA = σ3 ◦ φ2 . Lemma 2.3. Let X be an oval in Z2p . Then, the following holds. (1) |φ2 (X)| = 4. p+1 2 (2) |φ
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