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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 Wozencraft ensemble ： ウィキペディア英語版 Wozencraft ensemble In coding theory, the Wozencraft ensemble is a set of linear codes in which most of codes satisfy the Gilbert-Varshamov bound. It is named after John Wozencraft, who proved its existence. The ensemble is described by , who attributes it to Wozencraft. used the Wozencraft ensemble as the inner codes in his construction of strongly explicit asymptotically good code. ==Existence theorem== Theorem: Let $\backslash varepsilon$ > 0. For a large enough $k$, there exists an ensemble of inner codes $C\_^1,C\_^2,..,C\_^N$ of rate $\backslash frac$, where $N\; =\; q^k\; -\; 1$, such that for at least $\backslash left(\; \backslash right)N$ values of i, $C\_^i$ has relative distance $\backslash ge\; H\_q^\; (\backslash frac\; -\; \backslash varepsilon\; )$. Here relative distance is the ratio of minimum distance to block length. And $H\_q$ is the q-ary entropy function defined as follows: $H\_q(x)\; =\; xlog\_q(q-1)-xlog\_qx-(1-x)log\_q(1-x)$. In fact, to show the existence of this set of linear codes, we will specify this ensemble explicitly as follows: for $\backslash alpha\; \backslash in\; \backslash mathbb\_\; -\; \backslash$, the inner code $C\_^\backslash alpha\; :\backslash mathbb\_q^k\; \backslash to\; \backslash mathbb\_q^$, is defined as $C\_^\backslash alpha\; (x)\; =\; (x,\backslash alpha\; x)$. Here we can notice that $x\; \backslash in\; \backslash mathbb\_q^k$ and $\backslash alpha\; \backslash in\; \backslash mathbb\_$. We can do the multiplication $\backslash alpha\; x$ since $\backslash mathbb\_q^k$ is isomorphic to $\backslash mathbb\_$. This ensemble is due to Wozencraft and is called the Wozencraft ensemble. For any x and y in $\backslash mathbb\_q^k$, we have the following facts: # $C\_^\backslash alpha\; (x)\; +\; C\_^\backslash alpha\; (y)\; =\; (x,\backslash alpha\; x)+(y,\backslash alpha\; y)\; =\; (x\; +\; y,\backslash alpha\; (x\; +\; y))\; =\; C\_^\backslash alpha\; (x\; +\; y)$ # For any $a\; \backslash in\; F\_q$, $aC\_^\backslash alpha\; (x)\; =\; a(x,\backslash alpha\; x)\; =\; \backslash left(\; \backslash right)\; =\; C\_^\backslash alpha\; (ax)$ So $C\_^\backslash alpha$ is a linear code for every $\backslash alpha\; \backslash in\; \backslash mathbb\_\; -\; \backslash$. Now we know that Wozencraft ensemble contains linear codes with rate $\backslash frac$. In the following proof, we will show that there are at least $\backslash left(\; \backslash right)N$ those linear codes having the relative distance $\backslash ge\; H\_q^\; (\backslash frac\; -\; \backslash varepsilon\; )$, i.e. they meet the Gilbert-Varshamov bound. 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「Wozencraft ensemble」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.