On the classification of balanced binary sequences of period2^n-1(Corresp.)

Let
U
be the set of all binary sequences of period
p=2^{n}-1
containing
(p+1)/2
ones and
(p-1)/2
zeros per period. There is a lattice of interesting subsets of
U
, the smallest of which is the set
PN
(the maximum-length linear shift register sequences of period
p
). In between are sets with the run statistics of
PN
, with the correlation properties of
PN
, with the "span-
n
property" (that every nonzero subseqnonce of length
n
occurs in each period), and others. Results concerning the interrelationships of these subsets are obtained, examples are given to show that certain intersections of subsets are nonempty, and conjectures are formulated regarding other intersections of subsets. For example, it is conjectured that all span-
n
sequences with the two-level autocorrelation property are in class
PN
. Some relationships between run properties and correlation properties of binary sequences are also obtained.