On binary sequences with specific linear complexity and correlation properties.
For many applications, such as cryptography and digital communications, binary
sequences with certain specific properties are required. These include a balance
of 0's and 1's in a period, ideal runs frequencies, good auto- and cross-correlation
spectra, and high linear complexity.
Perfect Linear Complexity Profile sequences (PLCPs) have the linear complexity
of all subsequences (starting with the first bit) equal to half the length of the
subsequence (this is the expected value for a random sequence). We investigate
the density - proportion of ones - of finite length PLCPs, both in general and for
specific examples. We gain results on the average, maximal and minimal densities,
as well as their limits as the length tends to infinity. We also study whether the
PLCP property is preserved under various decimations.
PLCPs are characterised by a simple linear recurrence modulo 2. We look at
similar "nearly" perfect profiles and characterise sequences with these profiles in
terms of similar recurrences. Also sequences with a PLCP up to a point and then
constant complexity are characterised in terms of the convergents in the continued
fraction expansion of the generating function of PLCPs, and we look briefly at their
corresponding periods. Sequences with bounded jumps in their linear complexity
are discussed and a method of generating them is suggested.
The interleaving of shifts of a sequence with out-of-phase auto-correlation equal to
-1 and balance, in a specific order, seems to be a fundamental method of generating
longer sequences with this auto-correlation property. It is shown that two pairs of
families of these sequences, derived in different ways, are in fact equivalent. The
analysis highlights the general method mentioned above, and so provides examples
of families of sequences with 2-valued auto-correlation by changing the ingredients
in the interleaving pattern. We also look at the cross-correlation of sequences with
this interleaved structure.