IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 27, NO. 6, AUGUST 2009 965 Design of Rate-Compatible Structured LDPC Codes for Hybrid ARQ Applications Mostafa El-Khamy, Jilei Hou, and Naga Bhushan Abstract—In this paper, families of rate-compatible protograph-based LDPC codes that are suitable for incremental- redundancy hybrid ARQ applications are constructed. A systematic technique to construct low-rate base codes from a higher rate code is presented. The base codes are designed to be robust against erasures while having a good performance on error channels. A progressive node puncturing algorithm is devised to construct a family of higher rate codes from the base code. The performance of this puncturing algorithm is compared to other puncturing schemes. Using the techniques in this paper, one can construct a rate-compatible family of codes with rates ranging from 0.1 to 0.9 that are within 1 dB from the channel capacity and have good error floors. Index Terms—Low density parity check (LDPC) codes, hy- brid automatic repeat request (HARQ), progressive-edge growth (PEG), progressive puncturing, rate compatible error correcting codes, protographs, channel coding for wireless applications. I. I NTRODUCTION H YBRID automatic repeat request (HARQ) mechanisms are deployed in wireless systems (e.g. 3GPP HSPA, LTE) to improve their capacity and peak rates. HARQ mech- anisms can use forward error correcting codes to correct channel errors. Typically, in incremental redundancy HARQ mechanisms, the initial transmission consists of the data bits, encoded with an error detection code (e.g. CRC code) and a high rate forward error correcting code. If channel decoding of a transmission fails but a decoding error is detected, the receiver notifies the transmitter via a feedback channel and requests a retransmission via a negative acknowledgement (NAK) message. The subsequent retransmissions are not iden- tical, but consists of parity bits which, when appended at the receiver to the bits from the previous (re)transmissions, form a code with a lower coding rate. Thus incremental redundancy HARQ mechanisms require rate-compatible families of codes. A rate-compatible family of codes suitable for HARQ appli- cations is a nested family of codes with the same information block size but different coding rates. In such a rate-compatible family, the highest-rate code of the initial transmission should not only have a good performance alone, but should also have a good performance when the parity bits of the subsequent retransmissions are incrementally appended to it to form the lower-rate codes of the family. Manuscript received 30 September 2008; revised 15 March 2009. The material in this paper was presented in part at the International Symposium on Information Theory, Seattle 2006. The authors are with Corporate Research and Development, Qualcomm, San Diego (e-mail: mostafa@systems.caltech.edu, jhou@qualcomm.com and nbhush@qualcomm.com). Digital Object Identifier 10.1109/JSAC.2009.090814. Structured low density parity check (LDPC) codes have several advantages over non-structured LDPC codes due to their fast encoding and decoding structures and their natural scalability to different block sizes [1]–[3]. A structured LDPC code is characterized by a small graph called a protograph. A protograph G =(V , C , E ) is composed of a set of variable nodes V , a set of check nodes C and a set of edges E , such that a variable node in V can be connected to a check node in C via one or more edges in E [2]. The number of edges connected to a node is called the degree of the node. Each variable node, check node and edge in the protograph is said to be of a unique type.A lifted graph is constructed by making a number of copies of the protograph, determined by the desired information block size, and then carefully permuting the end-points of the edges between nodes of the same type in the protograph copies. It follows that the local neighborhood of any node in the lifted graph is the same as the local neighborhood of the corresponding node of the same type in the protograph. This facilitates faster optimization of structured LDPC codes by optimizing on the protograph level. Rate-compatible structured LDPC codes were previously considered by Divsalar et al. [4]. The information block size of these families is not constant and thus they are not suitable for HARQ applications. A rate-compatible family of codes with constant information block size can be constructed from a low-rate base code where the higher-rate codes in the family are constructed from the base code by puncturing parity bits such that bits which are transmitted in one code are also transmitted in all codes of lower rate in the family. Rate-compatible families of LDPC codes designed by code expurgation and lengthening, as proposed by Dolinar [5], can also have a constant information block size. However, rate-compatible families with constant information block size constructed by puncturing are more convenient and have the advantage of using a single decoder for all the codes in the family, where the decoder simply assumes 50% reliability for the punctured bits. Recently Ha et al. proposed a puncturing algorithm for LDPC codes with short block lengths [6]. Based on asymptotic analysis, optimizing the fraction of variable nodes to be punctured from each set of variable nodes in the base code with a given node degree has been considered by Ha et al. [7]. However, this approach is not directly applicable to structured LDPC codes since it is often the case that all the parity variable nodes are of the same degree (i.e. degree 2). By modeling punctured nodes as erasures, it was recently shown in [8] that for LDPC codes there exists a cutoff rate R c which depends on the degree distributions and the rate of the base code, such that through puncturing the base code, one could 0733-8716/09/$25.00 c  2009 IEEE