code

Let $A$ be an alphabet. A code over $A$ is any subset $C$ of the set of words $A^{*}$ on the alphabet $A$ such that $C$ has “uniquue factorization into letters,” i.e., such that for whenever $a_{1}\\ldots a_{n}=b_{1}\\ldots b_{m}$ , with all $a_{i},b_{j}\\in C$ , then we have $n=m$ and $a_{i}=b_{i}$ for all $i$ . In other words, every “word over $A$ ” generated by $C$ (considered as an alphabet) can be uniquely factored into “letters” in C.

An example of a subset of $A^{*}$ which is not a code is given by $C=\\{ab,c,a,bc\\}$ . Here the word $a b c$ can be written either as $(ab)c$ or as $a(bc)$ in terms of elements of $C$ . Since $ab\eq a$ nor $c\eq bc$ , $C$ is not a code.

If we fix a length $n$ for the words, i.e. we require that $C\\subset A^{n}$ , then we call $C$ a block code, and call $n$ the block length of the code. An important property of a code is the code’s minimum distance, given by the minimum Hamming distance between any pair of words in $C$ .

This notion of code is obviously very general. In practice (i.e., in coding theory) one typically takes codes with a little more structure. See, in particular, linear codes.

单词	Code
释义	code Let $A$ be an alphabet. A code over $A$ is any subset $C$ of the set of words $A^{}$ on the alphabet $A$ such that $C$ has “uniquue factorization into letters,” i.e., such that for whenever $a_{1}\\ldots a_{n}=b_{1}\\ldots b_{m}$ , with all $a_{i},b_{j}\\in C$ , then we have $n=m$ and $a_{i}=b_{i}$ for all $i$ . In other words, every “word over $A$ ” generated by $C$ (considered as an alphabet) can be uniquely factored into “letters” in C. An example of a subset of $A^{}$ which is not a code is given by $C=\\{ab,c,a,bc\\}$ . Here the word $a b c$ can be written either as $(ab)c$ or as $a(bc)$ in terms of elements of $C$ . Since $ab\eq a$ nor $c\eq bc$ , $C$ is not a code. If we fix a length $n$ for the words, i.e. we require that $C\\subset A^{n}$ , then we call $C$ a block code, and call $n$ the block length of the code. An important property of a code is the code’s minimum distance, given by the minimum Hamming distance between any pair of words in $C$ . This notion of code is obviously very general. In practice (i.e., in coding theory) one typically takes codes with a little more structure. See, in particular, linear codes.
随便看	NoCountableDenseSubsetOfACompleteMetricSpaceIsAGdelta no countable dense subset of a complete metric space is a G_δ NocyclesCondition no-cycles condition Noetherian noetherian NoetherianAndArtinianPropertiesAreInheritedInShortExactSequences Noetherian and Artinian properties are inherited in short exact sequences NoetherianModule Noetherian module NoetherianRing noetherian ring NoetherianTopologicalSpace Noetherian topological space NoetherNormalizationLemma Noether normalization lemma NonabelianGroup nonabelian group Nonahedron nonahedron NonassociativeAlgebra non-associative algebra NoncentralChisquaredRandomVariable non-central chi-squared random variable NoncommutativeDynamicModelingDiagrams