On the cycles and adjacencies in the complementary circulating register

A number theoretic view on binary shift registers

“…By ( 6), all elements in a cycle have the same weight, henceforth the weight of the cycle. These two observations together with (5) imply that for each cycle of weight w there exists a distinct cycle of weight n − w of the same length, with the exception of self-dual cycles, which might exist if n is even and w = n∕2 . Since such a (self-dual) cycle contains both x and x , we have 2 i x ≡ −x mod 2 n − 1 for some positive exponent i.…”

“…Next, by (7), edges can only exist between cycles whose weights differ by one. As a consequence, no intracyclic edge exists in PCR n [5]. A contradiction with respect to weights asserts that no extracyclic pairs exist either: On one hand, since x 2 < 2 n−1 , g c n (x 2 ) being on the same cycle as x 1 implies wt(x 1 ) = wt(g c n (x 2 )) = wt(x 2 ) + 1 .…”

“…Hence, there are at least (2 n +1) 2n cycles of length 2n. Hauge [5] proved that the length of each cycle is even and divides 2n with an odd quotient. We reformulate this as follows.…”

“…Magleby [8] and Fredricksen (as acknowledged in [5]) proved in different ways that the adjacency sets in PCR n have size at most 2. The number of adjacency sets of this maximal size was determined in [10,12] and later on in [5], each using a different method. We provide an alternative proof for both results.…”

Further Results on Pure Summing Registers and Complementary Ones

“…An alternate representation of C is to take the first digit of each state in order into a set, namely C = (a i a i+1 • • • a i+l−1 ). Two cycles are called to be adjacent if they share a conjugate or companion pair [4].…”