Nevertheless this doesn't touch the subject of proving the taps create a maximum-length LFSR. I don't believe that this is a complete set of requirements for taps. So the tap sequence has as its counterpart. If the tap sequence, in an n-bit LFSR, is, where the 0 corresponds to the x0 = 1 term, then the corresponding 'mirror' sequence is. Once one maximum-length tap sequence has been found, another automatically follows.There can be more than one maximum-length tap sequence for a given LFSR length.The set of taps must be relatively prime, and share no common divisor to all taps.The LFSR will only be maximum-length if the number of taps is even just 2 or 4 taps can suffice even for extremely long sequences.Wikipedia states the following on its LFSR page: Obviously, you can exhaustively check that it lands on every state except the zero state, but for large linear feedback shift registers (LFSR), this quickly becomes infeasible.
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