Sphere-packing
Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.
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 grid that packs the copies
Sampling a pattern on a grid makes copies of its frequencies, one for every point of a lattice, and the pattern can be read back from the samples only if the copies do not collide. For a pattern whose frequencies fill a disc the square grid wastes room between its copies. Searching every lattice of samples for the fewest that work finds a nearly hexagonal one every time — 124 samples where the square grid needs 169 — and the saving is the hexagonal packing's over the square, 13.4% in the limit. In three dimensions it is 29%, and in twenty-four the cubic grid needs 16,777,216 times as many samples as the best.
Named alongside it
The objects these essays reach for when they reach for this one.
Exhaustive searchAliasingBoundError-correcting codeFourier seriesFrequencyHamming distanceLatticePerfect codeRateSampling