Matpic
mathematics, in pictures
Concept

Coset — where it appears

One of the equal-sized shifted copies a subgroup cuts the whole group into.

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

Also named here as hamming code — the same set of essays touches all of them, so they are one junction rather than several.

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 received1011010H111010001110101101001010synsyndromethe bit it names101bit 1111bit 2110bit 3011bit 4100bit 5010bit 6001bit 7000no error1011010 gives syndrome 010, which is column 6 — so bit 6 is wrong and 1011000 is thecodewordthe syndrome is three bits and the message is four: the check finds the error without everrecovering what was sent

Finding the error without reading the message

Three parity checks on a seven-bit word produce three bits. If they are all zero nothing is wrong; otherwise they are the number of the position that broke. The message is never consulted, because the answer does not depend on it.

computation · error correcting codes

Named alongside it

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

Error correcting codeHamming codeCounting argumentDecision procedureHamming distanceLinearityMinimum distanceParityParity checkPerfect codeSphere packing boundSyndrome

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