On Undetected Error Probability of Binary Matrix Ensembles

In this paper, an analysis of the undetected error probability of ensembles of binary matrices is presented. The ensemble called the Bernoulli ensemble whose members are considered as matrices generated from i.i.d. Bernoulli source is mainly considered here. The main contributions of this work are (i) derivation of the error exponent of the average undetected error probability and (ii) closed form expressions for the variance of the undetected error probability. It is shown that the behavior of the exponent for a sparse ensemble is somewhat different from that for a dense ensemble. Furthermore, as a byproduct of the proof of the variance formula, simple covariance formula of the weight distribution is derived.Comment: 9 pages, a part of the paper was submitted to ISIT 200

Average Stopping Set Weight Distribution of Redundant Random Matrix Ensembles

In this paper, redundant random matrix ensembles (abbreviated as redundant random ensembles) are defined and their stopping set (SS) weight distributions are analyzed. A redundant random ensemble consists of a set of binary matrices with linearly dependent rows. These linearly dependent rows (redundant rows) significantly reduce the number of stopping sets of small size. An upper and lower bound on the average SS weight distribution of the redundant random ensembles are shown. From these bounds, the trade-off between the number of redundant rows (corresponding to decoding complexity of BP on BEC) and the critical exponent of the asymptotic growth rate of SS weight distribution (corresponding to decoding performance) can be derived. It is shown that, in some cases, a dense matrix with linearly dependent rows yields asymptotically (i.e., in the regime of small erasure probability) better performance than regular LDPC matrices with comparable parameters.Comment: 14 pages, 7 figures, Conference version to appear at the 2007 IEEE International Symposium on Information Theory, Nice, France, June 200

A Coding Theoretic Approach for Evaluating Accumulate Distribution on Minimum Cut Capacity of Weighted Random Graphs

The multicast capacity of a directed network is closely related to the $s$-$t$ maximum flow, which is equal to the $s$-$t$ minimum cut capacity due to the max-flow min-cut theorem. If the topology of a network (or link capacities) is dynamically changing or have stochastic nature, it is not so trivial to predict statistical properties on the maximum flow. In this paper, we present a coding theoretic approach for evaluating the accumulate distribution of the minimum cut capacity of weighted random graphs. The main feature of our approach is to utilize the correspondence between the cut space of a graph and a binary LDGM (low-density generator-matrix) code with column weight 2. The graph ensemble treated in the paper is a weighted version of Erd\H{o}s-R\'{e}nyi random graph ensemble. The main contribution of our work is a combinatorial lower bound for the accumulate distribution of the minimum cut capacity. From some computer experiments, it is observed that the lower bound derived here reflects the actual statistical behavior of the minimum cut capacity.Comment: 5 pages, 2 figures, submitted to IEEE ISIT 201