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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 Generalized minimum-distance decoding ： ウィキペディア英語版 Generalized minimum-distance decoding In coding theory, generalized minimum-distance (GMD) decoding provides an efficient algorithm for decoding concatenated codes, which is based on using an errors-and-erasures decoder for the outer code. A naive decoding algorithm for concatenated codes can not be an optimal way of decoding because it does not take into account the information that maximum likelihood decoding (MLD) gives. In other words, in the naive algorithm, inner received codewords are treated the same regardless of the difference between their hamming distances. Intuitively, the outer decoder should place higher confidence in symbols whose inner encodings are close to the received word. David Forney in 1966 devised a better algorithm called generalized minimum distance (GMD) decoding which makes use of those information better. This method is achieved by measuring confidence of each received codeword, and erasing symbols whose confidence is below a desired value. And GMD decoding algorithm was one of the first examples of soft-decision decoders. We will present three versions of the GMD decoding algorithm. The first two will be randomized algorithms while the last one will be a deterministic algorithm. ==Setup== # Hamming distance : Given two vectors $u,\; v\backslash in\backslash sum^n$ the Hamming distance between u and v, denoted by $\backslash Delta(u,\; v)$, is defined to be the number of positions in which u and v differ. # Minimum distance : Let $C\backslash subseteq\backslash sum^n$ be a code. The minimum distance of code C is defined to be $d\; =\; \backslash min$ where $c\_1\; \backslash ne\; c\_2\; \backslash in\; C$ # Code concatenation : Given $m\; =\; (m\_1,\; \backslash ldots,\; m\_K)\; \backslash in\; ()^K$, consider two codes which we call outer code and inner code $C\_\backslash text\; =\; ()^K\; \backslash rightarrow\; ()^N,\; C\_\backslash text\; :\; ()^k\; \backslash rightarrow\; ()^n$, and their distances are $D$ and $d$. A concatenated code can be achieved by $C\_\backslash text\; \backslash circ\; C\_\backslash text\; (m)\; =\; (C\_\backslash text\; (C\_\backslash text\; (m)\_1),\; \backslash ldots,\; C\_\backslash text\; (C\_\backslash text\; (m)\_N\; ))$ where $C\_\backslash text(m)\; =\; ((C\_\backslash text\; (m)\_1,\; \backslash ldots,\; (m)\_N\; ))$. Finally we will take $C\_\backslash text$ to be RS code, which has an errors and erasure decoder, and $K\; =\; O(\backslash log)$, which in turn implies that MLD on the inner code will be poly($N$) time. # Maximum likelihood decoding(MLD) : MLD is a decoding method for error correcting codes, which outputs the codeword closest to the received word in Hamming distance. The MLD function denoted by $D\_\; :\; \backslash sum^n\; \backslash rightarrow\; C$ is defined as follows. For every $y\backslash in\backslash sum\_n$, $D\_(y)\; =\; \backslash arg\; \backslash min\_\backslash Delta(c,\; y)$. # Probability density function : A probability distribution $\backslash Pr()$ on a sample space $S$ is a mapping from events of $S$ to real numbers such that $\backslash Pr()\; \backslash ge\; 0$ for any event $A$, $\backslash Pr()\; =\; 1$, and $\backslash Pr(\backslash cup\; B)\; =\; \backslash Pr()\; +\; \backslash Pr()$ for any two mutually exclusive events $A$ and $B$ # Expected value : The expected value of a discrete random variable $X$ is $\backslash mathbb\; =\; \backslash sum\_x\backslash Pr(=\; x)$. 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「Generalized minimum-distance decoding」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.