Sparse Generator Matrices Research Articles

A Network Linear Block Coding Approach to Selective Detect-and-Forward Multi-Way Relaying With Differential Modulation

This paper introduces a network linear block coding framework for multi-way relaying with differential MPSK modulation. We consider a system with $K$ user terminals and $L$ relays employing a selective detect-and-forward protocol. We show that such a system can be represented as a systematic $(K+L,K)$ linear block code. This framework allows the use of linear systematic block codes as network codes yielding power saving gains at user terminals. We analyse the theoretical performance of our scheme with optimal decoding, showing that our system can achieve a diversity order that equals to the minimum Hamming distance of the associated code. Then we consider a sub-optimal decoder based on log-domain belief propagation, and present numerical results for three 4-ary linear block codes and two binary LDPC codes as examples. Theoretical and simulation results demonstrate a significant performance gain of our scheme over uncoded transmission, even though some relays may be silent occasionally. We present a technique based on hard thresholding the received samples at the terminals, that makes the performance closer to the case when no relay is silent, without any significant increase in complexity. Furthermore, we also present guidelines for choosing suitable codes, and show that systematic block codes with sparse generator matrices are in particular desirable for our system.

On a Family of Circulant Matrices for Quasi-Cyclic Low-Density Generator Matrix Codes

We present a new class of sparse and easily invertible circulant matrices that can have a sparse inverse though not being permutation matrices. Their study is useful in the design of quasi-cyclic low-density generator matrix codes, that are able to join the inner structure of quasi-cyclic codes with sparse generator matrices, so limiting the number of elementary operations needed for encoding. Circulant matrices of the proposed class permit to hit both targets without resorting to identity or permutation matrices that may penalize the code minimum distance and often cause significant error floors.

Linear-Codes-Based Lossless Joint Source-Channel Coding for Multiple-Access Channels

A general lossless joint source-channel coding (JSCC) scheme based on linear codes and random interleavers for multiple-access channels (MACs) is presented and then analyzed in this paper. By the information-spectrum approach and the code-spectrum approach, it is shown that a linear code with a good joint spectrum can be used to establish limit-approaching lossless JSCC schemes for correlated general sources and general MACs, where the joint spectrum is a generalization of the input-output weight distribution. Some properties of linear codes with good joint spectra are investigated. A formula on the "distance" property of linear codes with good joint spectra is derived, based on which, it is further proved that, the rate of any systematic codes with good joint spectra cannot be larger than the reciprocal of the corresponding alphabet cardinality, and any sparse generator matrices cannot yield linear codes with good joint spectra. The problem of designing arbitrary rate coding schemes is also discussed. A novel idea called "generalized puncturing" is proposed, which makes it possible that one good low-rate linear code is enough for the design of coding schemes with multiple rates. Finally, various coding problems of MACs are reviewed in a unified framework established by the code-spectrum approach, under which, criteria and candidates of good linear codes in terms of spectrum requirements for such problems are clearly presented.

Flexible forward error correction codes with application to partial media data recovery

Conventionally, linear block codes designed for packet erasure correction are targeted to recover all the lost source packets per block, when the fraction of lost data is smaller than the redundancy overhead. However, these codes fail to recover any lost packets, if the number of erasures just exceeds the limit for full recovery capability, while it can still be beneficial to recover part of the symbols. In addition, common linear block codes are not well suited for unequal error protection, since different block codes with different rates must be allocated for each priority class separately. These two problems motivate the design of more flexible forward error correction (FEC) codes for media streaming applications. We first review the performance of short and long linear block codes. Long block codes generally offer better error correction capabilities, but at the price of higher complexity and larger coding delay. Short block codes can be more appropriate in media streaming applications that require smooth performance degradation when the channel loss rate increases. We study a new class of linear block codes using sparse generator matrices that permit to optimize the performance of short block codes for partial recovery of the lost packets. In addition, the proposed codes are extended to the design of unequal erasure protection solutions. Simulations of practical video streaming scenarios demonstrate that the flexible sparse codes offer a promising solution with interesting error correction capabilities and small variance in the residual loss rate. They typically represent an effective trade-off between short block codes with limited flexibility, and long block codes with delay penalties.