A handful of flip-flops and one XOR gate make a stream that looks random but repeats on a perfect schedule. Pick the right feedback taps and a 4-bit register visits all fifteen non-zero states before returning — the maximal-length sequence behind scramblers, GPS codes, and cheap noise. Slide the clock and watch it walk every state once.
A linear-feedback shift register of width 4 with taps at bits 4 and 1 realises the primitive polynomial x⁴+x+1. Each clock: XOR the tapped bits, shift right, feed the XOR into the top bit. Because the polynomial is primitive, the register cycles through every one of the 2⁴−1 = 15 non-zero states exactly once before repeating (0 is a fixed point it never enters). A fail-loud self-check throws unless the measured period is 15 and all 15 states are distinct — the maximal-length guarantee.
An LFSR is pseudo-random, not random — fully predictable once you know the taps and one state (why it is a SCRAMBLER, never a cipher on its own). The period-15 maximality holds only for primitive tap sets; a wrong choice gives shorter cycles.