Ultimately periodic sequence
A sequence over some set
satisfying the condition
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for all sufficiently large values of and some
is called ultimately periodic with period
; if this condition actually holds for all
,
is called periodic (with period
). The smallest number
among all periods of
is called the least period of
. The periods of
are precisely the multiples of
. Moreover, if
should be periodic for some period
, it is actually periodic with period
.
One may characterize the ultimately periodic sequences over some field by associating an arbitrary sequence
over
with the formal power series
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Then is ultimately periodic with period
if and only if
is a polynomial over
. Any ultimately periodic sequence over a field is a shift register sequence. The converse is not true in general, as the Fibonacci sequence over the rationals shows (cf. Shift register sequence). However, the ultimately periodic sequences over a Galois field are precisely the shift register sequences. Periodic sequences (in particular, binary ones) with good correlation properties are important in engineering applications (cf. Correlation property for sequences).
References
[a1] | D. Jungnickel, "Finite fields: Structure and arithmetics" , Bibliographisches Inst. Mannheim (1993) |
Ultimately periodic sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ultimately_periodic_sequence&oldid=15942