ar X iv : m at h / 06 05 10 4 v 1 [ m at h . C O ] 3 M ay 2 00 6 Asymptotics for the number of n - quasigroups of order 4

The asymptotic form of the number of n-quasigroups of order 4 is 3 n+1 2 2 n +1 (1 + o(1)). An algebraic system that consists of a set Σ of cardinality |Σ| = k and an n-ary operation f : Σ n → Σ uniquely invertible by each of its arguments is called an n-quasigroup of order k. The function f can also be referred to as an n-quasigroup of order k (see [Bel72]). The value table of an n-quasigroup of order k is called a Latin n-cube of dimension k (if n = 2, a Latin square). Furthermore, there is a one-to-one correspondence between the n-quasigroups and the distance 2 MDS codes of length n + 1. It is not difficult to show that for each n there exist only two n-quasigroups of order 2 and 3 · 2 n different n-quasigroups of order 3, which constitute one equivalence class. In this work we study properties of n-quasigroups of order 4 and derive the asymptotic representation 3 n+1 2 2 n +1 (1+ o(1)) for their number. The results of the research were announced in [KP01]. For k > 4, the asymptotic form of the number of n-quasigroups and even the asymptotic form of its logarithm remain unknown. In Sections 1–4 we give necessary definitions and statements on quaternary distance 2 MDS codes and double-codes (Section 1), linear double-codes (Section 2), n-quasigroups of order 4 (Section 3), semilinear n-quasigroups of order 4 (Section 4). In Section 5 we prove that almost all (as n → ∞) n-quasigroups of order 4 are semilinear and establish asymptotically tight bounds on their number. In addition to the main result, the following lemmas can be viewed as stand-alone results: Lemma 1 on a linear anti-layer in a double-MDS-code, Lemma 4 on a semilinear layer in an n-quasi-group, as well as Lemmas 2 and 3 on the decomposability of double-MDS-codes and n-quasigroups, proved in [Kro02, Kro05], and their Corollary 3. 1. MDS codes and double-codes Let Σ = {0, 1, 2, 3} and n be a natural number. In this paper we study subsets of Σ n and functions defined on Σ n that have some properties specified below. The elements of Σ n will be called vertices. Denote by [n] the set of natural numbers from 1 to n. Given ¯ y = (y 1 ,. .. , y n), we put …