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  "body": "\n## Definition\n\nLet $A$ be a [[finite set|0BBTH3YSZA]] with at least two elements, the alphabet, and let $n$ be a positive integer.\nFor words $x, y \\in A^n$, the Hamming distance $d(x, y)$ is the number of positions $i$ with $x_i \\ne y_i$.\nA code of length $n$ over $A$ is a [[subset|0BJR9QR08E]] $C \\subseteq A^n$ with at least two elements; its elements are the codewords.\nThe minimum distance of $C$ is $d(C) = \\min \\set{d(x, y) : x, y \\in C,\\ x \\ne y}$.\nFor an integer $t \\ge 0$, the code $C$ is $t$-error-correcting if for each word $y \\in A^n$ there is at most one codeword $c \\in C$ with $d(c, y) \\le t$.\n\n## Remarks\n\nThe code $C$ is $t$-error-correcting if and only if $d(C) \\ge 2t + 1$.\nOne direction uses the triangle inequality of the Hamming distance: if $d(c, y) \\le t$ and $d(c', y) \\le t$, then $d(c, c') \\le 2t$.\nExample: the binary repetition code $C = \\set{000, 111}$ has $d(C) = 3$, so it is $1$-error-correcting.\nA word with at most one $1$ is within distance $1$ of $000$ only, and a word with at least two $1$'s is within distance $1$ of $111$ only.\nNon-example: $C = \\set{000, 011}$ is not $1$-error-correcting, because $001$ is at distance $1$ from both codewords.\nA code with $M$ codewords, length $n$ and minimum distance $d$ is called an $(n, M, d)$ code.\nWhen $A$ is a finite field and $C$ is a subspace of $A^n$, the code is a linear code.\n",
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