منابع مشابه
n-Ary Quasigroups of Order 4
We characterize the set of all n-ary quasigroups of order 4: every n-ary quasigroup of order 4 is permutably reducible or semilinear. Permutable reducibility means that an n-ary quasigroup can be represented as a composition of k-ary and (n− k +1)-ary quasigroups for some k from 2 to n−1, where the order of arguments in the representation can differ from the original order. The set of semilinea...
متن کاملOn the number of n-ary quasigroups of finite order
LetQ.n; k/ be the number of n-ary quasigroups of order k. We derive a recurrent formula forQ.n; 4/. We prove that for all n 2 and k 5 the following inequalities hold: k 3 2 n=2 k 1 2 n=2 < log2Q.n; k/ ck.k 2/ ; where ck does not depend on n. So, the upper asymptotic bound for Q.n; k/ is improved for any k 5 and the lower bound is improved for odd k 7. This research was partially supported by th...
متن کاملOn reducibility of n-ary quasigroups
An n-ary operation Q : Σ → Σ is called an n-ary quasigroup of order |Σ| if in the equation x0 = Q(x1, . . . , xn) knowledge of any n elements of x0, . . . , xn uniquely specifies the remaining one. Q is permutably reducible if Q(x1, . . . , xn) = P ( R(xσ(1), . . . , xσ(k)), xσ(k+1), . . . , xσ(n) ) where P and R are (n− k+1)-ary and kary quasigroups, σ is a permutation, and 1 < k < n. Anm-ary ...
متن کاملOn reducibility of n-ary quasigroups, II
An n-ary operation Q : Σ → Σ is called an n-ary quasigroup of order |Σ| if in the equation x0 = Q(x1, . . . , xn) knowledge of any n elements of x0, . . . , xn uniquely specifies the remaining one. Q is permutably reducible ifQ(x1, . . . , xn) = P ` R(xσ(1), . . . , xσ(k)), xσ(k+1), . . . , xσ(n) ́ where P and R are (n − k + 1)-ary and k-ary quasigroups, σ is a permutation, and 1 < k < n. An m-a...
متن کاملOn decomposability of 4-ary distance 2 MDS codes, double-codes, and n-quasigroups of order 4
The union of two disjoint (n, 4n−1, 2) MDS codes in {0, 1, 2, 3}n is a double-MDS-code. If the adjacency graph of the double-MDS-code is not connected, then the double-code can be decomposed into double-MDS-codes of smaller lengths. If the graph has more than two connected components, then the MDS codes are also decomposable. The result has an interpretation in terms of n-quasigroups of order 4...
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ژورنال
عنوان ژورنال: SIAM Journal on Discrete Mathematics
سال: 2009
ISSN: 0895-4801,1095-7146
DOI: 10.1137/070697331