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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 Johnson bound ： ウィキペディア英語版 Johnson bound The Johnson bound is a limit on the size of error-correcting codes, as used in coding theory for data transmission or communications. == Definition == Let $C$ be a q-ary code of length $n$, i.e. a subset of $\backslash mathbb\_q^n$. Let $d$ be the minimum distance of $C$, i.e. :$d\; =\; \backslash min\_\; d(x,y)$ , where $d(x,y)$ is the Hamming distance between $x$ and $y$. Let $C\_q(n,d)$ be the set of all q-ary codes with length $n$ and minimum distance $d$ and let $C\_q(n,d,w)$ denote the set of codes in $C\_q(n,d)$ such that every element has exactly $w$ nonzero entries. Denote by $|C|$ the number of elements in $C$. Then, we define $A\_q(n,d)$ to be the largest size of a code with length $n$ and minimum distance $d$: :$A\_q(n,d)\; =\; \backslash max\_\; |C|.$ Similarly, we define $A\_q(n,d,w)$ to be the largest size of a code in $C\_q(n,d,w)$: :$A\_q(n,d,w)\; =\; \backslash max\_\; |C|.$ Theorem 1 (Johnson bound for $A\_q(n,d)$): If $d=2t+1$, :$A\_q(n,d)\; \backslash leq\; \backslash frac\; (q-1)^i\; +\; \backslash frac\; (q-1)^\; -\; A\_q(n,d,d)\}\; \}.$ If $d=2t$, :$A\_q(n,d)\; \backslash leq\; \backslash frac\; (q-1)^i\; +\; \backslash frac\; (q-1)^\; \}\; \}.$ Theorem 2 (Johnson bound for $A\_q(n,d,w)$): (i) If $d\; >\; 2w$, :$A\_q(n,d,w)\; =\; 1.$ (ii) If $d\; \backslash leq\; 2w$, then define the variable $e$ as follows. If $d$ is even, then define $e$ through the relation $d=2e$; if $d$ is odd, define $e$ through the relation $d\; =\; 2e\; -\; 1$. Let $q^$ * = q - 1. Then, :$A\_q(n,d,w)\; \backslash leq\; \backslash lfloor\; \backslash frac\; \backslash lfloor\; \backslash frac\; \backslash lfloor\; \backslash cdots\; \backslash lfloor\; \backslash frac\; \backslash rfloor\; \backslash cdots\; \backslash rfloor\; \backslash rfloor$ where $\backslash lfloor\; ~~\; \backslash rfloor$ is the floor function. Remark: Plugging the bound of Theorem 2 into the bound of Theorem 1 produces a numerical upper bound on $A\_q(n,d)$. 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「Johnson bound」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.