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Second-Order Weight Distributions

机译：二阶权重分布

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摘要

A fundamental property of codes, the second-order weight distribution, is proposed to solve the problems such as computing second moments of weight distributions of linear code ensembles. A series of results, parallel to those for weight distributions, is established for second-order weight distributions. In particular, an analogue of MacWilliams identities is proved. The second-order weight distributions of regular LDPC code ensembles are then computed. As easy consequences, the second moments of weight distributions of regular LDPC code ensembles are obtained. Furthermore, the application of second-order weight distributions in random coding approach is discussed. The second-order weight distributions of the ensembles generated by a so-called 2-good random generator or parity-check matrix are computed, where a 2-good random matrix is a kind of generalization of the uniformly distributed random matrix over a finite filed and is very useful for solving problems that involve pairwise or triple-wise properties of sequences. It is shown that the 2-good property is reflected in the second-order weight distribution, which thus plays a fundamental role in some well-known problems in coding theory and combinatorics. An example of linear intersecting codes is finally provided to illustrate this fact.
机译：提出了代码的基本属性，即二次权重分布，以解决诸如计算线性码集合的权重分布的第二矩之类的问题。对于二阶重量分布，建立了一系列与重量分布平行的结果。尤其证明了MacWilliams身份的类似物。然后计算常规LDPC码集合的二阶权重分布。作为简单的结果，获得了常规LDPC码集合的权重分布的第二矩。此外，还讨论了二阶权重分布在随机编码方法中的应用。计算由所谓的2良好随机生成器或奇偶校验矩阵生成的集合的二阶权重分布，其中2良好随机矩阵是有限域上均匀分布的随机矩阵的一种推广。对于解决涉及序列的成对或三向属性的问题非常有用。结果表明，二优属性反映在二阶权重分布中，因此在编码理论和组合学中的一些众所周知的问题中起着根本作用。最后提供了一个线性相交代码的例子来说明这一事实。

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