1 8 Ja n 20 07 On decomposability of 4 - ary distance 2 MDS codes , double - codes , and n - quasigroups of order 4 ∗
نویسنده
چکیده
A subset S of {0, 1, . . . , 2t − 1}n is called a t-fold MDS code if every line in each of n base directions contains exactly t elements of S. The adjacency graph of a t-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 t-fold MDS codes of smaller lengths. In the case t = 2, the theory has the following application. 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 doubleMDS-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 n-quasigroups of order 4.
منابع مشابه
On 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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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 ...
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