Math 2.0

Information and communication theory, circuits Definition

Error-correcting code

Builds on Finite setSubset

Definition

Let AA be a finite set with at least two elements, the alphabet, and let nn be a positive integer. For words x,y∈Anx, y \in A^n, the Hamming distance d(x,y)d(x, y) is the number of positions ii with xi≠yix_i \ne y_i. A code of length nn over AA is a subset C⊆AnC \subseteq A^n with at least two elements; its elements are the codewords. The minimum distance of CC is d(C)=min⁡{d(x,y):x,y∈C, x≠y}d(C) = \min \set{d(x, y) : x, y \in C,\ x \ne y}. For an integer t≥0t \ge 0, the code CC is tt-error-correcting if for each word y∈Any \in A^n there is at most one codeword c∈Cc \in C with d(c,y)≤td(c, y) \le t.

Remarks

The code CC is tt-error-correcting if and only if d(C)≥2t+1d(C) \ge 2t + 1. One direction uses the triangle inequality of the Hamming distance: if d(c,y)≤td(c, y) \le t and d(c′,y)≤td(c', y) \le t, then d(c,c′)≤2td(c, c') \le 2t. Example: the binary repetition code C={000,111}C = \set{000, 111} has d(C)=3d(C) = 3, so it is 11-error-correcting. A word with at most one 11 is within distance 11 of 000000 only, and a word with at least two 11's is within distance 11 of 111111 only. Non-example: C={000,011}C = \set{000, 011} is not 11-error-correcting, because 001001 is at distance 11 from both codewords. A code with MM codewords, length nn and minimum distance dd is called an (n,M,d)(n, M, d) code. When AA is a finite field and CC is a subspace of AnA^n, the code is a linear code.

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  1. Element definition
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  3. Subset concept
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  7. Ordered pair definition
  8. Existential quantifier definition
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  17. Surjective function definition
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  21. Error-correcting code definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Error_correction_code
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TagX122TKFP9G
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MSC94B25, 94B05
DefinesError-correcting code
UsesFinite set, Subset
Changed2 November 2026

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