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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Long code (mathematics) ： ウィキペディア英語版 Long code (mathematics) In theoretical computer science and coding theory, the long code is an error-correcting code that is locally decodable. Long codes have an extremely poor rate, but play a fundamental role in the theory of hardness of approximation. ==Definition== Let $f\_1,\backslash dots,f\_\; :\; \backslash ^k\backslash to\; \backslash$ for $k=\backslash log\; n$ be the list of ''all'' functions from $\backslash ^k\backslash to\backslash$. Then the long code encoding of a message $x\backslash in\backslash ^k$ is the string $f\_1(x)\backslash circ\; f\_2(x)\backslash circ\backslash dots\backslash circ\; f\_(x)$ where $\backslash circ$ denotes concatenation of strings. This string has length $2^n=2^$. The Walsh-Hadamard code is a subcode of the long code, and can be obtained by only using functions $f\_i$ that are linear functions when interpreted as functions $\backslash mathbb\; F\_2^k\backslash to\backslash mathbb\; F\_2$ on the finite field with two elements. Since there are only $2^k$ such functions, the block length of the Walsh-Hadamard code is $2^k$. An equivalent definition of the long code is as follows: The Long code encoding of $j\backslash in()$ is defined to be the truth table of the Boolean dictatorship function on the $j$th coordinate, i.e., the truth table of $f:\backslash ^n\backslash to\backslash$ with $f(x\_1,\backslash dots,x\_n)=x\_i$.〔Definition 7.3.1 in ( Limits of Approximation Algorithms: PCPs and Unique Games (DIMACS Tutorial Lecture Notes) )〕 Thus, the Long code encodes a $(\backslash log\; n)$-bit string as a $2^n$-bit string. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Long code (mathematics)」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.