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The Arithmetic Inside Every QR Code
A single closed world of 256 numbers repairs a scratched disc and encrypts your bank transfer — because in it, division never fails.
Tear a corner off a QR code — cover a fifth of it with your thumb, smear it, let a printer misfire across one edge — and it will very often still resolve, cleanly, to the same web address it always pointed to. This is not luck, and it is not sloppy tolerance on the part of the scanner. At its strongest error-correction setting, as much as a third of the little black squares in a code carry no message at all; they are redundancy, and that redundancy is computed inside a number system that most people who scan a boarding pass every week have never heard of and would not believe if described. It is a universe with exactly 256 numbers in it. Nothing lives outside it. You can add, subtract, multiply, and — this is the part that should not be possible — divide by any nonzero number, and never once land somewhere the universe doesn’t reach.
That closed, finite, self-referential little world is called GF(2⁸), the Galois field of 256 elements, and it is not a curiosity kept alive by algebraists for sentimental reasons. A finite field — a Galois field, in the older name — is a number system with only finitely many elements in which all four…
