Quasi-Cyclic LDPC Code Design for Block-Fading Channels Yueqian Li and Masoud Salehi Department of Electrical and Computer Engineering Northeastern University, Boston, MA 02115 Email: li.yu@neu.edu, salehi@ece.neu.edu Abstract—We investigate quasi-cyclic low-density parity-check (QC-LDPC) codes for nonergodic block-fading channels. A con- struction method for designing QC-LDPC codes is presented. With careful design, the proposed QC-LDPC codes exhibit the same good performance as their corresponding random root- LDPC codes introduced in [1] using iterative belief-propagation (BP) decoding. Moreover, the structure of the proposed QC- LDPC codes makes an efficient encoding method possible. Index Terms—Block-fading channels, efficient encoding, quasi- cyclic low-density parity-check (QC-LDPC) codes. I. I NTRODUCTION The nonergodic block-fading channel was first studied in [2] and [3]. This channel model is appropriate for mobile com- munication systems affected by slow fading. We consider the situation that a codeword with finite length can be transmitted only over a few independent fading blocks. Codes designed for block-fading channels are expected to employ the limited diversity that the channel provides and offer good coding gain. Recently, Boutros et al. proposed a family of low-density parity-check (LDPC) codes called root-LDPC codes for non- ergodic block-fading channels [1]. The root-LDPC codes are shown to achieve full diversity on block-fading channels and perform close to the outage limit when decoded using the iter- ative belief-propagation (BP) decoding. In the bipartite graph representation of root-LDPC codes, a subset of connections are deterministically selected to guarantee full diversity for information bits and the rest of connections are generated randomly. LDPC codes with randomly generated parity check matri- ces generally have good performance, but they lack enough structure to facilitate efficient encoding methods. For practical applications, it is suggested to use structured LDPC codes, which is the main motivation of this research. In this paper, instead of using random matrices, we construct parity check matrices of root-LDPC codes by tiling circulant matrices, i.e., by designing quasi-cyclic low-density parity-check (QC- LDPC) codes. The generator matrices of QC-LDPC codes are in systematic-circulant (SC) form with the requirement that the parity check matrices are full rank. We will describe how to construct full rank parity check matrices for QC-LDPC codes. The memory cost for encoding of QC-LDPC codes is greatly reduced and the encoder can be implemented using simple shift registers [4]. With proper design, the QC-LDPC codes can perform as good as randomly generated root-LDPC codes over block-fading channels. The rest of the paper is organized as follows. Section II introduces the channel model and establishes notations. In Section III, we first review the design of root-LDPC codes and then introduce the construction method of QC-LDPC codes for block-fading channels. We also discuss how to design QC-LDPC codes with large girths. Simulation results of the proposed QC-LDPC codes are presented in Section IV. In Section V, the encoding of QC-LDPC codes using generator matrix in SC form is discussed. Conclusions are finally drawn in Section VI. II. THE BLOCK- FADING CHANNEL MODEL AND NOTATION We consider a communication channel with an independent fading coefficient on each block. A codeword of length N can span over F fading blocks. Assuming BPSK symbols x l ∈ {+1, −1} are transmitted, the lth received signal is given by y l = α f x l + z l , l =1, 2 ··· N and f =1, 2 ··· F (1) where α f is the real Rayleigh fading coefficient on the f th fading block with E[|α f | 2 ]=1 and f = ⌈F · l/N ⌉ with ⌈r⌉ denoting the ceiling of r. Each noise sample is distributed according to z l ∼N (0,σ 2 ), where σ 2 = N 0 /2. The signal- to-noise ratio (SNR) is expressed as E b /N 0 , where E b is the energy per information bit. A simple example of a codeword in a block-fading channel with F =2 fading blocks is illustrated in Fig. 1. 1 2 A codeword of length N Fig. 1: A codeword of length N spans over F =2 fading blocks. The Shannon capacity of the block-fading channel is zero. Instead, the channel is characterized by its information outage limit, which is defined as P out = Pr(I (E b /N 0 , {α f }) <R), (2) where I (E b /N 0 , {α f }) is the instantaneous mutual informa- tion between input and output of the channel. R = K/N