We consider irregular product codes.In this class of codes, each codeword is represented by a matrix. The entries in each row (column) of the matrix should come from a component row (column) code. As opposed to (standard) product codes, we do not require that all component row codes nor all component column codes be the same. As we will see, relaxing this requirement can provide some additional attractive features including 1) allowing some regions of the codeword be more error-resilient 2) allowing a more refined spectrum of rates for finite-lengths and improved performance in some of these rates 3) more interaction between row and column codes during decoding. We study these codes over erasure channels. We find that for any $0 < ε< 1$, for many rate distributions on component row codes, there is a matching rate distribution on component column codes such that an irregular product code based on MDS codes with those rate distributions on the component codes has asymptotic rate $1 - ε$ and can decode on erasure channels (of alphabet size equal the alphabet size of the component MDS codes) with erasure probability $< ε$.
Spatially-coupled (SC) LDPC codes have recently emerged as an excellent choice for error correction in modern data storage and communication systems due to their outstanding performance. It has long been known that irregular graph codes offer performance advantage over their regular counterparts. In this paper, we present a novel combinatorial framework for designing finite-length irregular SC LDPC codes. Our irregular SC codes have the desirable properties of regular SC codes thanks to their structure while offering significant performance benefits that come with the node degree irregularity.…
Recently, it was found that clipping can significantly improve the section error rate (SER) performance of sparse regression (SR) codes if an optimal clipping threshold is chosen. In this paper, we propose irregularly clipped SR codes, where multiple clipping thresholds are applied to symbols according to a distribution, to further improve the SER performance of SR codes. Orthogonal approximate message passing (OAMP) algorithm is used for decoding. Using state evolution, the distribution of irregular clipping thresholds is optimized to minimize the SER of OAMP decoding. As a result, optimized …
In this paper, we consider the update bandwidth in distributed storage systems~(DSSs). The update bandwidth, which measures the transmission efficiency of the update process in DSSs, is defined as the total amount of data symbols transferred in the network when the data symbols stored in a node are updated. This paper contains the following contributions. First, we establish the closed-form expression of the minimum update bandwidth attainable by irregular array codes. Second, after defining a class of irregular array codes, called Minimum Update Bandwidth~(MUB) codes, which achieve the minimu…
Asymptotic Average Multiplicity of Structures within Different Categories of Trapping Sets, Absorbing Sets and Stopping Sets in Random Regular and Irregular LDPC Code Ensembles
Structured codes based on lattices were shown to provide enlarged capacity for multi-user communication networks. In this paper, we study capacity-approaching irregular repeat accumulate (IRA) codes over integer rings $\mathbb{Z}_{2^{m}}$ for $2^m$-PAM signaling, $m=1,2,\cdots$. Such codes feature the property that the integer sum of $K$ codewords belongs to the extended codebook (or lattice) w.r.t. the base code. With it, \emph{% structured binning} can be utilized and the gains promised in lattice based network information theory can be materialized in practice. In designing IRA ring codes, …
The huge amount of data produced in the fifth-generation (5G) networks not only brings new challenges to the reliability and efficiency of mobile devices but also drives rapid development of new storage techniques. With the benefits of fast access speed and high reliability, NAND flash memory has become a promising storage solution for the 5G networks. In this paper, we investigate a protograph-coded bit-interleaved coded modulation with iterative detection and decoding (BICM-ID) utilizing irregular mapping (IM) in the multi-level-cell (MLC) NAND flash-memory systems. First, we propose an enha…
This paper considers the problem of Quantitative Group Testing (QGT) where there are some defective items among a large population of $N$ items. We consider the scenario in which each item is defective with probability $K/N$, independently from the other items. In the QGT problem, the goal is to identify all or a sufficiently large fraction of the defective items by testing groups of items, with the minimum possible number of tests. In particular, the outcome of each test is a non-negative integer which indicates the number of defective items in the tested group. In this work, we propose a non…
In this paper, we study the connection between polar codes and product codes. Our analysis shows that the product of two polar codes is again a polar code, and we provide guidelines to compute its frozen set on the basis of the frozen sets of the component polar codes. Moreover, we show how polar codes can be described as irregular product codes. We propose a two-step decoder for long polar codes taking advantage of this dual nature to heavily reduce decoding latency. Finally, we show that the proposed decoding technique outperforms both standard polar codes and state-of-the-art codes for opti…
