The Cyclic Redundancy Check

A parity bit notices one flipped bit. Real wires do not flip one bit. They deliver bursts — a relay clicks, a motor starts, lightning strikes three counties away — and forty consecutive bits arrive as noise. Parity catches that only half the time, because half of all bursts contain an even number of flips. Wesley Peterson and Daniel Brown published the fix in 1961, and it is a piece of algebra rather than a piece of engineering. Treat the message as a polynomial whose coefficients are its bits, working over the field with two elements — where one plus one is zero, so addition and subtraction are both exclusive-or and there are no carries anywhere. Divide by a fixed generator polynomial and append the remainder. The receiver divides again; a clean frame leaves nothing. What falls out of that is not a probability but a set of GUARANTEES. A generator of degree r detects every single-bit error, every odd number of errors if the polynomial is chosen well, and — the one that matters — EVERY burst of r bits or fewer. Not usually. Every one. Beyond r bits the guarantee stops and the miss rate steps to about two to the minus r, which for the 32-bit CRC in every Ethernet frame is one in four thousand million. It corrects nothing. That is the trade against the Hamming code this blueprint links to: Hamming repairs a single bit and scales badly as blocks grow, while a CRC only ever says yes or no, costs 32 bits whether the frame is 64 bytes or 1500, and catches the burst that Hamming would not even notice. FULLY BUILDABLE, and the proof is the point: inject every possible burst of every length and count what slips through. The wall sits exactly where the algebra says it does.