Second-Order Weight Distributions
arXiv:1007.3108 · doi:10.1109/TIT.2011.2162272
Abstract
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.
10 pages, accepted for publication in IEEE Transactions on Information Theory, May 2011
References in corpus (5)
- On the Asymptotic Weight and Stopping Set Distribution of Regular LDPC Ensembles
- Weight Distributions of Regular Low-Density Parity-Check Codes over Finite Fields
- Linear-Codes-Based Lossless Joint Source-Channel Coding for Multiple-Access Channels
- Good Random Matrices over Finite Fields
- Growth Rate of the Weight Distribution of Doubly-Generalized LDPC Codes: General Case and Efficient Evaluation