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On the decomposistion of orthogonal arrays
, 1999
"... When an orthogonal array is projected on a small number of factors, as is done in screening experiments, the question of interest is the structure of the projected design, by which we mean its decomposition in terms of smaller arrays of the same strength. In this paper we investigate the decompositi ..."
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When an orthogonal array is projected on a small number of factors, as is done in screening experiments, the question of interest is the structure of the projected design, by which we mean its decomposition in terms of smaller arrays of the same strength. In this paper we investigate the decomposition of arrays of strength t having t + 1 factors. The decomposition problem is wellunderstood for symmetric arrays on s = 2 symbols. In this paper we derive some general results on decomposition, with particular attention to arrays on s = 3 symbols. We give a new proof of the regularity of arrays of index 1 when s = 2 or 3, and show by counterexample that the result doesn’t extend to larger s. For s = 3 we also construct an indecomposable array of index 2. Finally, we determine the structure of completely decomposable arrays on 3 symbols having strength 2 and index 2, 3 or 4. array. Key words. Orthogonal array, decomposition, projection, simple orthogonal array, regular orthogonal
On the degree of local permutation polynomials∗
"... Every Latin square of prime or prime power order s corresponds to a polynomial in 2 variables over the finite field on s elements, called the local permutation polynomial. What characterizes this polynomial is that its restrictions to one variable are permutations. We discuss the general form of lo ..."
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Every Latin square of prime or prime power order s corresponds to a polynomial in 2 variables over the finite field on s elements, called the local permutation polynomial. What characterizes this polynomial is that its restrictions to one variable are permutations. We discuss the general form of local permutation polynomials and prove that their total degree is at most 2s−4, and that this bound is sharp. We also show that the degree of the local permutation polynomial for Latin squares having a particular form is at most s − 2. This implies that circulant Latin squares of prime order p correspond to local permutation polynomials having degree at most p − 2. Finally, we discuss a special case of circulant Latin squares whose local permutation polynomial is linear in both variables.