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  "body": "\n## Summary\n\nAn error-correcting code is a set of words of fixed length, the codewords, chosen far apart from each other in Hamming distance.\nA message is sent as a codeword; if a few symbols are corrupted on the way, the receiver still finds the nearest codeword.\nA code whose codewords are at distance at least $2t + 1$ from each other corrects any $t$ errors.\nCodes link combinatorics with algebra: good codes come from [[block designs|4JCSS2Z1BA]], [[Hadamard matrices|J0N4J606X0]] and finite fields.\nThe node [[error-correcting code|X122TKFP9G]] gives the precise definition.\n\n## Explanations\n\n### L1 human\n\nSuppose you send one bit, but noise may flip it.\nSend it three times instead: $000$ for $0$ and $111$ for $1$.\nIf the receiver gets $010$, the most likely message is $0$, since only one bit had to flip.\nThis is the repetition code; it corrects one error at the price of sending three times as much.\nBetter codes, such as the Hamming code, correct errors at a much lower price.\n",
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