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The affinity of a permutation of a finite vector space
For a permutation f of an n-dimensional vector space V over a finite field of order q we let k -affinity ( f ) denote the number of k-flats X of V such that f ( X ) is also a k-flat. By k -spectrum ( n , q ) we mean the set of integers k -affinity ( f ) , where f runs through all permutations of V....
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| Published in: | Finite fields and their applications 2007, Vol.13 (1), p.80-112 |
|---|---|
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites Items that cite this one |
| Online Access: | Get full text |
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| Summary: | For a permutation
f of an
n-dimensional vector space
V over a finite field of order
q we let
k
-affinity
(
f
)
denote the number of
k-flats
X of
V such that
f
(
X
)
is also a
k-flat. By
k
-spectrum
(
n
,
q
)
we mean the set of integers
k
-affinity
(
f
)
, where
f runs through all permutations of
V. The problem of the complete determination of
k
-spectrum
(
n
,
q
)
seems very difficult except for small or special values of the parameters. However, we are able to determine
(
n
-
1
)
-spectrum
(
n
,
2
)
and establish that
0
∈
k
-spectrum
(
n
,
q
)
in the following cases: (i)
q
≥
3
and
1
≤
k
≤
n
-
1
; (ii)
q
=
2
,
3
≤
k
≤
n
-
1
; (iii)
q
=
2
,
k
=
2
,
n
≥
3
odd. For
1
≤
k
≤
n
-
1
and
(
q
,
k
)
≠
(
2
,
1
)
, the maximum of
k
-affinity
(
f
)
is obtained when
f is any semiaffine mapping. We conjecture that the next to largest value of
k
-affinity
(
f
)
occurs when
f is a transposition, and we are able to prove it when
q
=
2
,
k
=
2
,
n
≥
3
and when
q
≥
3
,
k
=
1
,
n
≥
2
. |
|---|---|
| ISSN: | 1071-5797 1090-2465 |
| DOI: | 10.1016/j.ffa.2005.07.004 |