If this interests you, make sure to look into Golay codes: https://en.wikipedia.org/wiki/Binary_Golay_code https://en.wikipedia.org/wiki/Binary_Golay_code This

If this interests you, make sure to look into Golay codes: https://en.wikipedia.org/wiki/Binary_Golay_code https://en.wikipedia.org/wiki/Binary_Golay_code This code is a mathematical miracle. Apparently, number theory comes together exactly such as to allow you to code 12 bits into 23 bits. Such that you can correct all possible 3 bit errors, but none of the possible 4 bit errors (which means it is efficient). AND it detects all of the possible 7 bit errors, but none of the possible 8 bit errors. That is very efficient. So in exchange for storing all your data about twice, you can detect all errors where ~30% of the bits are flipped. You can even _correct_ all errors where only 12% of the bits were flipped! I still consider it magic that such an efficient and 'perfect' code exists. There is unfortunately a proof that is the best one, and that there is no bigger perfect one.