A Protograph-Based Design of Quasi-Cyclic Spatially Coupled LDPC Codes Li Chen †, Shiyuan Mo †, Daniel J. Costello, Jr. ‡, David G. M. Mitchell §, Roxana Smarandache ‡ † School of Electronics and Information Technology, Sun Yat-sen University, Guangzhou, China ‡ Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN, USA § Klipsch School of Electrical and Computer Engineering, New Mexico State University, Las Cruces, NM, USA Emails: chenli55@mail.sysu.edu.cn, moshiy@mail2.sysu.edu.cn, {costello.2, rsmarand}@nd.edu, dgmm@nmsu.edu Abstract—Spatially coupled (SC) low-density parity-check (LDPC) codes can achieve capacity approaching performance with low message recovery latency when using sliding window (SW) decoding. An SC-LDPC code constructed from a pro- tograph can be generated by first coupling a chain of block protographs and then lifting the coupled protograph using per- mutation matrices. This paper introduces a systematic design of SC-LDPC codes to eliminate 4-cycles in the coupled protograph. Using a quasi-cyclic (QC) lifting, we obtain QC-SC-LDPC codes of girth at least eight. Coupling a chain of block protographs implies spreading edges from one protograph to the others. Our protograph-based design can be viewed as guiding the edge spreading and also the graph-lifting process. Simulation results show the design leads to improved decoding performance, particularly in the error floor, compared to random designs. Index Terms—Cycles, LDPC codes, protographs, spatially coupled codes, sliding window decoding. I. I NTRODUCTION Since the original work of Thorpe [1], it has been recog- nized that protographs provide an efficient method of con- structing low-density parity-check (LDPC) codes. Analyzing the iterative decoding thresholds and minimum distance prop- erties of small protographs sheds light on constructing code ensembles with good asymptotic properties by applying a graph-lifting procedure [2]. If the permutation matrices used in the lifting procedure are circulants (shifted identity matrices), a quasi-cyclic (QC) ensemble results, a desirable property for practical implementation. Another important aspect of code design is to maximize the girth of the Tanner graph. For protograph-based constructions of QC-LDPC codes, this can be accomplished by applying the Fossorier condition [3] to the graph-lifting. The protograph-based method has also been used to construct good spatially coupled LDPC (SC-LDPC) codes [4]. An edge-spreading procedure is first applied to a chain of block protographs in order to introduce memory. This results in a two-step code design procedure, first the edge spreading and then the graph-lifting, to achieve good asymptotic properties and a large girth, respectively. Several constructions for QC-SC-LDPC codes have been proposed in [5]–[8]. In this paper, we take a new protograph- based systematic design approach to insure large girth for QC- SC-LDPC codes. The idea depends on the fact that the girth of a lifted graph is lower bounded by the girth of its base graph. Hence, starting from a block protograph with good asymptotic properties, we design the edge spreading in two stages to maximize the girth and minimize the number of short cycles in the SC protograph. Then, in the lifting phase, we use circulants to further improve the girth by applying the Fossorier condition. Besides improving our ability to find protographs with large girth and a small number of short cycles, the two-stage approach makes it easier to apply the Fossorier condition, since the SC protograph has already been designed to achieve these objectives. The edge-spreading procedure can be interpreted as decom- posing a base matrix B (corresponding to a block protograph) into a number of submatrices, which are used to form an SC base matrix B SC . In our approach, we identify several compo- nent blocks of B SC that guide the design of the submatrices, leading to an SC protograph with good girth properties. By further performing a graph-lifting of B SC using the Fossorier condition to generate an SC parity-check matrix H SC , we show that it is possible to achieve a girth of at least eight. Simulation results show that substantial performance gains, particularly in the error floor, are achieved using the two-stage design approach compared to random designs. II. SC-LDPC CODES The construction of a protograph-based SC-LDPC code can be described as a two-step procedure – first protograph coupling and then lifting [4]. A protograph [1] is a small bipartite graph with n c check nodes and n v variable nodes, where n c <n v . It can be represented by a base matrix B =[B(r, s)] nc×nv , (1) where B(r, s) is the row-r column-s entry, 1 ≤ r ≤ n c and 1 ≤ s ≤ n v . The entries represent the number of edges that connect check node r to variable node s in the protograph. For example, Fig. 1(a) shows a protograph defined by B = [3 3]. We first replicate the protograph as an infinite chain as shown in Fig. 1(b), then spread edges from the variable nodes of the protograph at time instant t by connecting them to check nodes at time instants t+1 to t+ω. Replicating this spreading over all protographs yields an SC protograph with coupling width ω, as shown in Fig. 1(c). This edge spreading can be interpreted as decomposing B into ω +1 submatrices of the same size, i.e., B 0 , B 1 ,..., B ω , such that B(r, s)= ω  i=0 B i (r, s), (2) 2017 IEEE International Symposium on Information Theory (ISIT) 978-1-5090-4096-4/17/$31.00 ©2017 IEEE 1683