Abstract
We present an application of generalized strong complete mappings to construction of a family of mutually orthogonal Latin squares. We also determine a cycle structure of such mapping which form a complete family of MOLS. Many constructions of generalized strong complete mappings over an extension of finite field are provided.
Highlights
A strong complete mapping is a complete mapping which is an orthomorphism. These mappings are used for a construction of Knut Vic designs and they exist only for the groups of order n where gcd(n, 6) = 1
Our point of interest is an application of these mappings to construction of mutually orthogonal Latin squares (MOLS)
If θ generates a complete set of MOLS over a finite Abelian group of order n as in the Theorem 2.1, θ has either one fixed element and one cycle of the length n or one fixed element and two cycles of the length n−1 2
Summary
A strong complete mapping is a complete mapping which is an orthomorphism. These mappings are used for a construction of Knut Vic designs and they exist only for the groups of order n where gcd(n, 6) = 1. Let Fq be a finite field of order q. We consider complete and strong complete mappings (and orthomorphisms) over (Fq(x), +). In [1], strong complete mappings over finite fields are called very complete mappings. Our point of interest is an application of these mappings to construction of mutually orthogonal Latin squares (MOLS). Many constructions of such mappings over finite fields will be presented
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