Linear binary shift register circuits utilizing a minimum number of mod-2 adders

A family of linear binary shift register circuits is described in terms of generator polynomials
g(x)
which require fewer mod-
2
adders than those derived from the companion matrices. These circuits are derived from various combinations of the functions
a(x), b(x)
, and
x^{d}
in
g(x) = 1 + a(x)x^{d}b(x)
and
c(D), e(D)
, and
D^{m}
of the reciprocal polynomial
g^{\ast}(x) = f(D) = 1 + c(D)D^{m}e(D)
. Tables are provided for all simplifications thus achievable for primitive polynomial of degrees
5
through
9
, and for some nonprimitive polynomials.