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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 Binary Goppa code ： ウィキペディア英語版 Binary Goppa code In mathematics and computer science, the binary Goppa code is an error-correcting code that belongs to the class of general Goppa codes originally described by Valerii Denisovich Goppa, but the binary structure gives it several mathematical advantages over non-binary variants, also providing a better fit for common usage in computers and telecommunication. Binary Goppa codes have interesting properties suitable for cryptography in McEliece-like cryptosystems and similar setups. ==Construction and properties== A binary Goppa code is defined by a polynomial $g(x)$ of degree $t$ over a finite field $GF(2^m)$ without multiple zeros, and a sequence $L$ of $n$ distinct elements from $GF(2^m)$ that aren't roots of the polynomial: : $\backslash forall\; i,j\; \backslash in\; \backslash :\; L\_i\; \backslash in\; GF(2^m)\; \backslash and\; L\_i\; \backslash neq\; L\_j\; \backslash and\; g(L\_i)\; \backslash neq\; 0$ Codewords belong to the kernel of syndrome function, forming a subspace of $\backslash ^n$: : $\backslash Gamma(g,L)=\backslash left\backslash ^\; \backslash frac\; \backslash equiv\; 0\; \backslash mod\; g(x)\; \backslash right\backslash \}$ Code defined by a tuple $(g,L)$ has minimum distance $2t+1$, thus it can correct $t=\backslash left\backslash lfloor\; \backslash frac\; \backslash right\backslash rfloor$ errors in a word of size $n-mt$ using codewords of size $n$. It also possesses a convenient parity-check matrix $H$ in form : $$ H=VD=\begin 1 & 1 & 1 & \cdots & 1\\ L_0^1 & L_1^1 & L_2^1 & \cdots & L_^1\\ L_0^2 & L_1^2 & L_2^2 & \cdots & L_^2\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ L_0^t & L_1^t & L_2^t & \cdots & L_^t \end \begin \frac & & & & \\ & \frac & & & \\ & & \frac & & \\ & & & \ddots & \\ & & & & \frac \end Note that this form of the parity-check matrix, being composed of a Vandermonde matrix $V$ and diagonal matrix $D$, shares the form with check matrices of alternant codes, thus alternant decoders can be used on this form. Such decoders usually provide only limited error-correcting capability (in most cases $t/2$). For practical purposes, parity-check matrix of a binary Goppa code is usually converted to a more computer-friendly binary form by a trace construction, that converts the $t$-by-$n$ matrix over $GF(2^m)$ to a $mt$-by-$n$ binary matrix by writing polynomial cofficients of $GF(2^m)$ elements on $m$ successive rows. 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「Binary Goppa 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.