Minimum distance
Named by 3 essays across one field — 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 polynomial through the gaps
Write the message as the coefficients of a polynomial and send its values instead. Any k of them determine the polynomial, so it does not matter which ones are lost — and it does not matter how many, as long as k survive.
Named alongside it
The objects these essays reach for when they reach for this one.
Error-correcting codeHamming distanceSphere-packing boundCheck digitCode rateCosetCounting argumentErasureFinite fieldHamming codeHypercubeInterpolation