Matpic
mathematics, in pictures
Concept

Sphere packing bound

The limit on how many codewords a code can hold, being the number of words divided by the size of one correction ball.

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

0000010100111001011101112 codewords in the 3-cube, minimum distance 3 — 1 error corrected, 2 detectedthe 2 balls of radius 1 hold 4 words each and cover all 8 exactly once: the code is perfect

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 four data bits, three parity bitsdataparityweight00000000000101130010110300111014010011140101100301100013011101041000101310011104101001141011000311000103110100141110100411111117sixteen codewords of length seven, minimum distance 3 — measured over all 120pairsweights: 1×0 · 7×3 · 7×4 · 1×7, and the lightest non-zero one weighs 3

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.

computation · error correcting codes

Named alongside it

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

Error correcting codeHamming distanceMinimum distanceCheck digitCode rateCosetCounting argumentHamming codeHypercubeMetricModular arithmeticParity

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