When *Voyager* visited Saturn and Jupiter, data for the pictures used blocks of Golay code. Each 24-bit block of data could have up to three errors, and the computers here on Earth could fix these errors. The Golay code is thus an error-correcting code. It was originally published in 1949 with Marcel Golay's half-page paper, "Notes on Digital Coding". Today, this paper is considered one of the most remarkable papers ever published, with deep, deep connections to group theory, graph theory, number theory, combinatorics, game theory, multidimensional geometry, and even particle physics.

The 23-bit Golay code is called a

*perfect* code. Any integer from 0 to

is within distance three of one of the code words, so that up to three errors can be detected and corrected. The automorphism group is the Mathieu group

The only other nontrivial perfect codes are the ternary Golay code and the Hamming code. The code words of weight 7 are elements of an

(4, 7, 23) Steiner system.

The 24-bit Golay code is called a

*semiperfect* code. Any integer from 0 to

is within distance four of one of the code words. Up to four errors can be detected and up to three errors can be corrected. The automorphism group is the Mathieu group

The code words of weight eight are elements of an

(5, 8, 24) Steiner system.