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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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