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首页> 外文期刊>IEEE Transactions on Information Theory >Entropy/length profiles, bounds on the minimal covering of bipartite graphs, and trellis complexity of nonlinear codes
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Entropy/length profiles, bounds on the minimal covering of bipartite graphs, and trellis complexity of nonlinear codes

机译：熵/长度分布图，二分图的最小覆盖范围的边界以及非线性代码的网格复杂度

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摘要

The trellis representation of nonlinear codes is studied from a new perspective. We introduce the new concept of entropy/length profile (ELP). This profile can be considered as an extension of the dimension/length profile (DLP) to nonlinear codes. This elaboration of the DLP, the entropy/length profiles, appears to be suitable to the analysis of nonlinear codes. Additionally and independently, we use well-known information-theoretic measures to derive novel bounds on the minimal covering of a bipartite graph by complete subgraphs. We use these bounds in conjunction with the ELP notion to derive both lower and upper bounds on the state complexity and branch complexity profiles of (nonlinear) block codes represented by any trellis diagram. We lay down no restrictions on the trellis structure, and we do not confine the scope of our results to proper or one-to-one trellises only. The basic lower bound on the state complexity profile implies that the state complexity at any given level cannot be smaller than the mutual information between the past and the future portions of the code at this level under a uniform distribution of the codewords. We also devise a different probabilistic model to prove that the minimum achievable state complexity over all possible trellises is not larger than the maximum value of the above mutual information over all possible probability distributions of the codewords. This approach is pursued further to derive similar bounds on the branch complexity profile. To the best of our knowledge, the proposed upper bounds are the only upper bounds that address nonlinear codes. The novel lower bounds are tighter than the existing bounds. The new quantities and bounds reduce to well-known results when applied to linear codes.
机译：从一个新的角度研究了非线性代码的网格表示。我们介绍了熵/长度分布（ELP）的新概念。该轮廓可以看作是尺寸/长度轮廓（DLP）对非线性代码的扩展。 DLP（熵/长度分布图）的详细说明似乎适合于非线性代码的分析。此外，我们独立地使用众所周知的信息理论方法，通过完整的子图在二部图的最小覆盖率上得出新的界。我们将这些界限与ELP概念结合使用，以得出由任何网格图表示的（非线性）分组码的状态复杂度和分支复杂度分布图的上限和下限。我们不对网格结构设置任何限制，也没有将结果范围限制为适当的或一对一的网格。状态复杂度分布图的基本下限意味着，在给定级别的状态下，在代码字的均匀分布下，状态复杂度不能小于该级别的代码的过去和将来部分之间的相互信息。我们还设计了一个不同的概率模型，以证明在所有可能的网格上可实现的最小状态复杂度不大于在码字的所有可能的概率分布上上述互信息的最大值。进一步追求该方法，以得出分支复杂度轮廓上的相似边界。据我们所知，提出的上限是唯一可解决非线性代码的上限。新颖的下界比现有界限更严格。当应用于线性代码时，新的数量和界限会减少到众所周知的结果。

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