Hamming distance
Named by 15 essays across 5 fields — each of them below, with the objects they name alongside it.
Distance is a picture
A message is a corner of a cube and an error is a step along an edge. Everything a code can do is decided by how far apart the corners it uses are — and that is a fact about a drawing.
![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.
A walk that changes one thing at a time
Counting from nothing to fifteen in binary changes four digits at once somewhere in the middle. There is another order through the same sixteen words in which every step changes exactly one — and it is a closed walk on a four-dimensional cube.
Circles that are diamonds and squares
The theorem hands over a formula for distance. Take the formula as a definition, change the exponent in it, and the set of points one unit from the origin stops being round — while remaining, in every sense that matters, a circle.
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.
![How many codewords lie within each radius, for a [7,3] code over 11 symbols. A bar for each decoding radius, its height the largest number of codewords found inside a ball of that radius around a randomly drawn received word, with the unique-decoding radius and the Johnson radius marked.](/f/how-many-codewords-lie-within-each-radius-for-a-7-3-code-e1fb4777.svg)
Past half the distance
A code of minimum distance five corrects two errors, and every account stops there. Two is the largest number for which the answer is unique — and a decoder that returns a short list instead of one answer reaches considerably further, which can be measured by counting the codewords in a ball.
The nearest consistent verdict
When a court's majorities contradict each other, one repair is to announce the consistent verdict that disagrees with the judges least. It treats the premises and the conclusion alike, which neither of the two standard procedures does. On the classic case it returns a three-way tie; on five judges, with every question weighted equally, it never returns a single answer on a troubled profile at all — and what breaks the tie is a decision about which question matters more.
The rate a noisy channel allows
A channel that flips one bit in ten can still carry messages with as few errors as anyone likes — at up to 0.531 message bits per transmitted bit, and at no rate above that. The number is Shannon's capacity, 1 − H(p). Repetition reaches reliability only by sending nothing; a code chosen at random gets there at any rate below the limit; and the reason there is a limit at all is a count of how many flip patterns a block of noise can hold.
Every place changes back
A closed walk through every corner of a cube changes one place at each step, and each place, having changed, must change back before the walk returns home. So every place changes an even number of times — which is why no walk on three places can share the work evenly, why perfect sharing is possible only when the number of places is a power of two, and what sorts the 1,344 walks on the 4-cube into exactly four kinds.
Orthogonal squares are a code
Write down each cell of a set of orthogonal Latin squares as a word — its row, its column, and its entry in each square — and no two words agree in more than one place. That is not a pleasant accident of the squares. It is exactly what being Latin and being orthogonal say, it makes the list an error-correcting code as good as any code of its size can be, and the squares a field builds turn out to be a Reed–Solomon code, the one on every compact disc.
A walk that beats trying everything
To decide whether clauses of three letters can all be satisfied, the obvious method tries all 2ⁿ assignments. Uwe Schöning's method, from 1999, starts at a random assignment and wanders: pick a clause that is false, flip one of its letters at random, and repeat three times as many times as there are letters. A single try usually fails, but it succeeds with chance at least about (3/4)ⁿ, so about (4/3)ⁿ tries are enough — and the reason is a walk on a line that goes the wrong way two times in three.
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.
How few distances fix a word
An unknown word of a thousand bits can be read one bit at a time with a thousand questions. Ask instead for its distance from a chosen word, and each answer is worth about half of log₂ n bits, so roughly two hundred distances fix it. A balance reading many coins at once is the same question. So is Mastermind, in which Knuth's rule finds any of the 1,296 codes in at most five guesses and a mean of 4.476, and no first guess is best on both counts.
Parallel planes take a vote
Draw a four-dimensional space with two values to a coordinate as a 4 × 4 grid, and mark its hyperplanes: thirty-two patterns any two of which differ on eight of sixteen squares. A polynomial's top coefficient can be read off any plane, and parallel planes do not overlap, so an error spoils only one vote. One of them flew to Mars, and in 2016 it turned out to reach Shannon's limit.
Wrong guesses that arrive together
Seven players in red or blue hats each see every hat but their own, and must guess their own colour or pass. Every guess is a coin toss, yet with the Hamming code the team wins seven times in eight — not by making guesses better, but by making the wrong ones arrive together. The best chance for any number of players is fixed by the fewest losses that cover the cube.
Named alongside it
The objects these essays reach for when they reach for this one.
Error-correcting codeExhaustive searchHypercubeCounting argumentPerfect codeHamming codeParityBinaryBoundChannel capacityEntropyFinite field