These things come up from time to time - does anyone know if there's an
algorithm for selecting taps for the maximum-length sequence, or is it
just by-guess-and-by-gosh?
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Depending on the period, there's generally more than just one
maximal-length sequence, but I'm not aware of any algorithm which
can successfully predict which sequences will be maximal length.
Google "Table of irreducible polynomials" for some fascinating
reading, if you like that kind of thing.
I do
I have a photocopy of Marsh's "TABLE OF IRREDUCIBLE POLYNOMIALS OVER
GF(2) THROUGH DEGREE 19" which stops at degree 14, and the
introductory material indicates that all of the sequences were
unveiled/tested through the use of software.
That is, I guess, EXORing all of the taps sequentially and examining
the output sequences.