Matpic
mathematics, in pictures
Concept

Check digit

An extra digit appended to an identifying number, computed from the others so that common copying mistakes give a number that fails the check. Which errors a scheme catches depends on the arithmetic used to compute it.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

A code on the 3-cube, and the balls around its words. The corners of a hypercube with the chosen codewords marked and the words within one error of each shaded.

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.

computation · Error-correcting codes No pair of weights modulo ten catches everything. Two 10 × 10 grids over pairs of alternating weights modulo ten, shaded by the share of single errors and of swaps each catches. No pair is full in both; weights 3 and 1, ringed, catch every single error and 88.9% of swaps.

Ten digits need a symmetry that does not commute

A check digit should catch one wrong digit and two neighbours swapped. Over eleven symbols a weighted sum does both; over the ten decimal digits no weighted sum can, no scheme of any shape built on adding modulo ten can, and the reason is the same as the reason Euler's thirty-six officers cannot be paraded. What works is the ten symmetries of a pentagon, which do not commute.

computation · Error-correcting codes

Named alongside it

The objects these essays reach for when they reach for this one.

Error-correcting codeModular arithmeticCode rateCommutativityDihedral groupExhaustive searchHamming distanceHypercubeImpossibilityLatin squareMetricMinimum distance

All concepts