5 research outputs found

    On decomposability of 4-ary distance 2 MDS codes, double-codes, and n-quasigroups of order 4

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    A subset SS of {0,1,...,2tβˆ’1}n\{0,1,...,2t-1\}^n is called a tt-fold MDS code if every line in each of nn base directions contains exactly tt elements of SS. The adjacency graph of a tt-fold MDS code is not connected if and only if the characteristic function of the code is the repetition-free sum of the characteristic functions of tt-fold MDS codes of smaller lengths. In the case t=2t=2, the theory has the following application. The union of two disjoint (n,4nβˆ’1,2)(n,4^{n-1},2) MDS codes in {0,1,2,3}n\{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 as a test for reducibility of nn-quasigroups of order 4. Keywords: MDS codes, n-quasigroups, decomposability, reducibility, frequency hypercubes, latin hypercubesComment: 19 pages. V2: revised, general case q=2t is added. Submitted to Discr. Mat

    On reducibility of n-ary quasigroups

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    An nn-ary operation Q:Snβˆ’>SQ:S^n -> S is called an nn-ary quasigroup of order ∣S∣|S| if in the equation x0=Q(x1,...,xn)x_{0}=Q(x_1,...,x_n) knowledge of any nn elements of x0x_0, ..., xnx_n uniquely specifies the remaining one. QQ is permutably reducible if Q(x1,...,xn)=P(R(xs(1),...,xs(k)),xs(k+1),...,xs(n))Q(x_1,...,x_n)=P(R(x_{s(1)},...,x_{s(k)}),x_{s(k+1)},...,x_{s(n)}) where PP and RR are (nβˆ’k+1)(n-k+1)-ary and kk-ary quasigroups, ss is a permutation, and 1<k<n1<k<n. An mm-ary quasigroup SS is called a retract of QQ if it can be obtained from QQ or one of its inverses by fixing nβˆ’m>0n-m>0 arguments. We prove that if the maximum arity of a permutably irreducible retract of an nn-ary quasigroup QQ belongs to {3,...,nβˆ’3}\{3,...,n-3\}, then QQ is permutably reducible. Keywords: n-ary quasigroups, retracts, reducibility, distance 2 MDS codes, latin hypercubesComment: 13 pages; presented at ACCT'2004 v2: revised; bibliography updated; 2 appendixe

    Asymptotics for the number of n-quasigroups of order 4

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    The asymptotic form of the number of n-quasigroups of order 4 is 3n+122n+1(1+o(1))3^{n+1} 2^{2^n +1} (1+o(1)). Keywords: n-quasigroups, MDS codes, decomposability, reducibility.Comment: 15 p., 3 fi
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