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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 Rank error-correcting code ： ウィキペディア英語版 Rank error-correcting code In coding theory, rank codes (also called Gabidulin codes) are non-binary〔Codes for which each input symbol is from a set of size greater than 2.〕 linear error-correcting codes over not Hamming but ''rank'' metric. They described a systematic way of building codes that could detect and correct multiple random ''rank'' errors. By adding redundancy with coding ''k''-symbol word to a ''n''-symbol word, a rank code can correct any errors of rank up to ''t'' = ⌊ (''d'' − 1) / 2 ⌋, where ''d'' is a code distance. As an erasure code, it can correct up to ''d'' − 1 known erasures. A rank code is an algebraic linear code over the finite field $GF(q^N)$ similar to Reed–Solomon code. The rank of the vector over $GF(q^N)$ is the maximum number of linearly independent components over $GF(q)$. The rank distance between two vectors over $GF(q^N)$ is the rank of the difference of these vectors. The rank code corrects all errors with rank of the error vector not greater than ''t''. == Rank metric == Let $X^n$ — ''n''-dimensional vector space over the finite field $GF\backslash left(\; \backslash right)$, where $q$ is a power of a prime, $N$ is an integer and $\backslash left(u\_1,\; u\_2,\; \backslash dots,\; u\_N\backslash right)$ with $u\_i\; \backslash in\; GF(q)$ is a base of the vector space over the field $GF\backslash left(\; \backslash right)$. Every element $x\_i\; \backslash in\; GF\backslash left(\; \backslash right)$ can be represented as $x\_i\; =\; a\_u\_1\; +\; a\_u\_2\; +\; \backslash dots\; +\; a\_u\_N$. Hence, every vector $\backslash vec\; x\; =\; \backslash left(\; \backslash right)$ over $GF\backslash left(\; \backslash right)$ can be written as matrix: : $$ \vec x = \left\| c} a_ & a_ & \ldots & a_ \\ a_ & a_ & \ldots & a_ \\ \ldots & \ldots & \ldots & \ldots \\ a_ & a_ & \ldots & a_ \end} \right\| ''Rank of the vector'' $\backslash vec\; x$ over the field $GF\backslash left(\; \backslash right)$ is a rank of the corresponding matrix $A\backslash left(\; \backslash right)$ over the field $GF\backslash left(\; \backslash right)$ denoted by $r\backslash left(\; \backslash right)$. The set of all vectors $\backslash vec\; x$ is a space $X^n\; =\; A\_N^n$. The map $\backslash vec\; x\; \backslash to\; r\backslash left(\; \backslash vec\; x;\; q\; \backslash right)$) defines a norm over $X^n$ and a ''rank metric'': : $$ d\left( \right) = r\left( \right) 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「Rank error-correcting 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.