A class of algebraically structured quasi-cyclic (QC) low-density parity-check (LDPC) codes and their convolutional counterparts is presented. The QC codes are described by sparse parity-check matrices comprised of blocks of circulant matrices. The sparse parity-check representation allows for practical graph-based iterative message-passing decoding. Based on the algebraic structure, bounds on the girth and minimum distance of the codes are found, and several possible encoding techniques are described. The performance of the QC LDPC block codes compares favorably with that of randomly constructed LDPC codes for short to moderate block lengths. The performance of the LDPC convolutional codes is superior to that of the QC codes on which they are based; this performance is the limiting performance obtained by increasing the circulant size of the base QC code. Finally, a continuous decoding procedure for the LDPC convolutional codes is described. Index Terms-Circulant matrices, iterative decoding, low-density parity-check (LDPC) block codes, LDPC convolutional codes, LDPC codes, message-passing, quasi-cyclic (QC) codes. I. INTRODUCTION L OW-density parity-check (LDPC) codes have attracted considerable attention in the coding community because they can achieve near-capacity performance with iterative message-passing decoding and sufficiently long block sizes. For example, in [1], Chung et al. presented a block length (ten million bits) rate-LDPC code that achieves reliable performance-a bit error rate (BER)-on an additive white Gaussian noise (AWGN) channel with a signal-to-noise ratio (SNR) within 0.04 dB of the Shannon limit. For many practical applications, however, the design of good codes with shorter block lengths is desired. Moreover,

Abstract-The goal of this paper is to establish which practical routing schemes for wireless networks are most suitable for power-limited and bandwidth-limited communication regimes. We regard channel state information (CSI) at the receiver and point-to-point capacity-achieving codes for the additive white Gaussian noise (AWGN) channel as practical features, interference cancellation (IC) as possible, but less practical, and synchronous cooperation (CSI at the transmitters) as impractical. We consider a communication network with a single source node, a single destination node, and 1 intermediate nodes placed equidistantly on a line between them. We analyze the minimum total transmit power needed to achieve a desired end-to-end rate for several schemes and demonstrate that multihop communication with spatial reuse performs very well in the power-limited regime, even without IC. However, within a class of schemes not performing IC, single-hop transmission (directly from source to destination) is more suitable for the bandwidth-limited regime, especially when higher spectral efficiencies are required. At such higher spectral efficiencies, the gap between single-hop and multihop can be closed by employing IC, and we present a scheme based upon backward decoding that can remove all interference from the multihop system with an arbitrarily small rate loss. This new scheme is also used to demonstrate that rates of (log ) are achievable over linear wireless networks even without synchronous cooperation.

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-We consider a wireless communication system with a single source node, a single destination node, and multiple relay nodes placed equidistantly between them. We limit our analysis to the case of coded TDMA multihop transmission, i.e., the nodes do not cooperate and do not try to access the channel simultaneously. Given a global constraint on bandwidth, we determine the number of hops that achieves the desired end-to-end rate with the least transmission power. Furthermore, we examine how the optimum hop number changes when an end-toend delay con-straint is introduced using the spherepacking bound and computer simulations. The analysis demonstrates that the optimum number of hops depends on the end-to-end spectral efficiency and the path-loss exponent. Specifically, we show the existence of an asymptotic per-link spectral efficiency, which is the most preferable spectral efficiency in TDMA multihop transmission.

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