Perfect code
Named by 3 essays across one field — each of them below, with the objects they name alongside it.
![The sixteen words of the [7,4] Hamming code. A table of sixteen seven-bit codewords with their data bits, parity bits and weights.](/f/the-sixteen-words-of-the-7-4-hamming-code-03f0d582.svg)
Sixteen spheres that fill a cube
A hundred and twenty-eight seven-bit words, sixteen of them chosen, and a ball of eight around each. Sixteen times eight is a hundred and twenty-eight exactly — so the balls tile the space with nothing left over, and the code wastes nothing at all.
The best a code can be
A code is a set of words chosen far apart, and every construction answers "here is one" rather than "here is the best". The best can be computed at small lengths, and putting four classical bounds beside the exact answer shows which of them is doing the work and where none of them is.
A code built from an icosahedron
Write down which corners of an icosahedron are not joined by an edge, put the table beside a diagonal of ones, and add the twelve rows together in every combination. The 4,096 strings of 24 bits that come out differ from each other in at least eight places, occur in only five weights, and — with one bit deleted — fill the space of 23-bit strings so exactly that every one of the 8,388,608 lies within three flips of exactly one of them. This is the Golay code, which carried Voyager's pictures home, and its words of weight eight cover every five of the twenty-four positions exactly once.
Named alongside it
The objects these essays reach for when they reach for this one.
Error-correcting codeHamming distanceExhaustive searchBoundCosetCounting argumentGolay codeHamming codeIcosahedronMinimum distanceRateSphere-packing