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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 Reed–Muller codes ： ウィキペディア英語版 Reed–Muller code Reed–Muller codes are a family of linear error-correcting codes used in communications. Reed–Muller codes belong to the classes of locally testable codes and locally decodable codes, which is why they are useful in the design of probabilistically checkable proofs in computational complexity theory. They are named after Irving S. Reed and David E. Muller. Muller discovered the codes, and Reed proposed the majority logic decoding for the first time. Special cases of Reed–Muller codes include the Hadamard code, the Walsh–Hadamard code, and the Reed–Solomon code. Reed–Muller codes are listed as RM(''d'', ''r''), where ''d'' is the order of the code, and ''r'' determines the length of code, ''n'' = 2 ''r''. RM codes are related to binary functions on the field GF(2 ''r'') over the elements . RM(0, ''r'') codes are repetition codes of length ''n'' = 2 ''r'', rate $\}$ and minimum distance ''d''min = ''n''. RM(1, ''r'') codes are parity check codes of length ''n'' = 2 ''r'', rate $R=\backslash tfrac$ and minimum distance $d\_\backslash min\; =\; \backslash tfrac$. RM(''r'' − 1, ''r'') codes are parity check codes of length ''n'' = 2 ''r''. RM(''r'' − 2, ''r'') codes are the family of extended Hamming codes of length ''n'' = 2 ''r'' with minimum distance ''d''min = 4.〔Trellis and Turbo Coding, C. Schlegel & L. Perez, Wiley Interscience, 2004, p149.〕 ==Construction== A generator matrix for a Reed–Muller code of length ''n'' = 2''d'' can be constructed as follows. Let us write: :$X\; =\; \backslash mathbb\_2^d\; =\; \backslash .$ Note that each member of the set X is a point in $\backslash mathbb\_2^d$. We define in n-dimensional space $\backslash mathbb\_2^n$ the indicator vectors :$\backslash mathbb\_A\; \backslash in\; \backslash mathbb\_2^n$ on subsets $A\; \backslash subset\; X$ by: : together with, also in $\backslash mathbb\_2^n$, the binary operation :$w\; \backslash wedge\; z\; =\; (w\_1\; \backslash cdot\; z\_1,\; \backslash ldots\; ,\; w\_n\; \backslash cdot\; z\_n\; ),$ referred to as the ''wedge product'' (this wedge product is not to be confused with the wedge product defined in exterior algebra). Here, $w=(w\_1,w\_2,\backslash ldots,w\_n)$ and $z=(z\_1,z\_2,\backslash ldots,\; z\_n)$ are points in $\backslash mathbb\_2^n$, and the operation $\backslash cdot$ is the usual multiplication in the field $\backslash mathbb\_2$. $\backslash mathbb\_2^d$ is a ''d''-dimensional vector space over the field $\backslash mathbb\_2$, so it is possible to write $(\backslash mathbb\_2)^d\; =\; \backslash$ We define in ''n''-dimensional space $\backslash mathbb\_2^n$ the following vectors with length ''n'': ''v''0 = (1, 1, 1, 1, 1, 1, 1, 1) and :$v\_i\; =\; \backslash mathbb\_$ where the ''H''''i'' are hyperplanes in $(\backslash mathbb\_2)^d$ (with dimension d −1): :$H\_i\; =\; \backslash$ 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「Reed–Muller code」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.