Analysis and Design of Serially Concatenated LDGM Codes

In this paper, we first present the asymptotic performance of serially concatenated low-density generator-matrix (SCLDGM) codes for binary input additive white Gaussian noise channels using discretized density evolution (DDE). We then provide a necessary condition for the successful decoding of these codes. The error-floor analysis along with the lower bound formulas for both LDGM and SCLDGM codes are also provided and verified. We further show that by concatenating inner LDGM codes with a high-rate outer LDPC code instead of concatenating two LDGM codes as in SCLDGM codes, good codes without error floors can be constructed. Finally, with an efficient DDE-based optimization approach that utilizes the necessary condition for the successful decoding, we construct optimized SCLDGM codes that approach the Shannon limit. The improved performance of our optimized SCLDGM codes is demonstrated through both asymptotic and simulation results.Comment: This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessibl

Capacity Achieving Linear Codes with Random Binary Sparse Generating Matrices

In this paper, we prove the existence of capacity achieving linear codes with random binary sparse generating matrices. The results on the existence of capacity achieving linear codes in the literature are limited to the random binary codes with equal probability generating matrix elements and sparse parity-check matrices. Moreover, the codes with sparse generating matrices reported in the literature are not proved to be capacity achieving. As opposed to the existing results in the literature, which are based on optimal maximum a posteriori decoders, the proposed approach is based on a different decoder and consequently is suboptimal. We also demonstrate an interesting trade-off between the sparsity of the generating matrix and the error exponent (a constant which determines how exponentially fast the probability of error decays as block length tends to infinity). An interesting observation is that for small block sizes, less sparse generating matrices have better performances while for large blok sizes, the performance of the random generating matrices become independent of the sparsity. Moreover, we prove the existence of capacity achieving linear codes with a given (arbitrarily low) density of ones on rows of the generating matrix. In addition to proving the existence of capacity achieving sparse codes, an important conclusion of our paper is that for a sufficiently large code length, no search is necessary in practice to find a deterministic matrix by proving that any arbitrarily selected sequence of sparse generating matrices is capacity achieving with high probability. The focus in this paper is on the binary symmetric and binary erasure channels.her discrete memory-less symmetric channels.Comment: Submitted to IEEE transaction on Information Theor

Coding for Crowdsourced Classification with XOR Queries

This paper models the crowdsourced labeling/classification problem as a sparsely encoded source coding problem, where each query answer, regarded as a code bit, is the XOR of a small number of labels, as source information bits. In this paper we leverage the connections between this problem and well-studied codes with sparse representations for the channel coding problem to provide querying schemes with almost optimal number of queries, each of which involving only a constant number of labels. We also extend this scenario to the case where some workers can be unresponsive. For this case, we propose querying schemes where each query involves only log n items, where n is the total number of items to be labeled. Furthermore, we consider classification of two correlated labeling systems and provide two-stage querying schemes with almost optimal number of queries each involving a constant number of labels.Comment: 6 page