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The affinity of a permutation of a finite vector space. (English) Zbl 1110.11036
Let \(f\) be a permutation of the \(n\)-dimensional vector space \(\mathbb F_q^n\) over a finite field \(\mathbb F_q\), then \(k\)-affinity(\(f\)) denotes the number of \(k\)-dimensional affine subspaces \(X\) of \(\mathbb F_q^n\) such that \(f(X)\) is again a \(k\)-dimensional affine subspace of \(\mathbb F_q^n\). The \(k\)-spectrum(\(n,q\)) is then the set of values \(k\)-affinity(\(f\)) for all permutations \(f\) of \(\mathbb F_q^n\). The authors show that, with few exceptions, \(0 \in k\)-spectrum(\(n,q\)) and present results on the largest values contained in \(k\)-spectrum(\(n,q\)). The paper is a continuation of X. D. Hou [Discrete Appl. Math. 154, 313–325 (2006; Zbl 1089.94020)].
MSC:
11T99 Finite fields and commutative rings (number-theoretic aspects)
05A05 Permutations, words, matrices
12E10 Special polynomials in general fields
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References:
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