On systematic design of universally capacity approaching rate-compatible sequences of LDPC code ensembles over binary-input output-symmetric memoryless channels Hamid Saeedi † , Hossein Pishro-Nik † , and Amir H. Banihashemi ⋄ † Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, MA, USA ⋄ Department of Systems and Computer Engineering, Carleton University, Ottawa, ON, Canada {hsaeedi,pishro@ecs.umass.edu; ahashemi@sce.carleton.ca} Abstract—Despite tremendous amount of research on the design of Low-Density Parity-Check (LDPC) codes with be- lief propagation decoding over different types of Binary-Input Output-Symmetric Memoryless (BIOSM) channels, most results on this topic are based on numerical methods and optimization which do not provide much insight into the design process. In particular, systematic design of provably capacity achieving sequences of LDPC code ensembles over the general class of BIOSM channels, has remained a fundamental open problem. For the case of the Binary Erasure channel, explicit construction of capacity achieving sequences have been proposed based on a property called the flatness condition. In this paper, we propose a systematic method to design universally capacity approaching rate-compatible LDPC code ensemble sequences over BIOSM channels. This is achieved by interpreting the flatness condition over the BEC, as a Successive Maximization (SM) principle that is generalized to other BIOSM channels to design a sequence of capacity approaching ensembles called the parent sequence. The SM principle is then applied to each ensemble within the parent sequence, this time to design rate-compatible puncturing schemes. As part of our results, we extend the stability condition which was previously derived for degree-2 variable nodes to other variable node degrees as well as to the case of rate-compatible codes. Consequently, we rigorously prove that using the SM principle, one is able to design universally capacity achieving rate-compatible LDPC code ensemble sequences over the BEC. Unlike the previous results on such schemes over the BEC in the literature, the proposed SM approach is naturally extendable to other BIOSM channels. The performance of the rate-compatible schemes designed based on our systematic method is comparable to those designed by optimization. I. I NTRODUCTION Low-Density Parity-Check (LDPC) codes have received much attention in the past decade. During this period there have been great achievements in the area of designing LDPC code ensembles with Belief Propagation (BP) decoding which exhibit an asymptotic performance practically close to the capacity over different types of channels, including the gen- eral class of Binary-Input Output-Symmetric Memoryless (BIOSM) channels [1]-[10]. In particular, for the Binary Era- sure Channel (BEC), the performance analysis and code design This work was supported by National Science Foundation under grants ECS- 0636569 and CCF- 0830614. have been addressed in both the asymptotic regime [3]-[8] and for finite block lengths [1], [2]. In [3], [4], [5], Shokrollahi et al. proposed a scheme to design sequences of LDPC code ensembles over the BEC, whose performance is proved to achieve the capacity for sufficiently large average check and variable node degrees. A more general category of capacity achieving sequences over the BEC were proposed in [11], [12], [13]. Construction and analysis of capacity achieving ensemble sequences of codes defined on graphs has also been studied in [6], [7], [8] for the BEC. A sequence of degree distributions with rate R is said to be capacity achieving over the BEC if the thresholds of the ensembles can be made arbitrarily close to 1 − R, the capacity upper bound over the BEC, as the average check and variable node degrees tend to infinity. For BIOSM channels, it is easier to consider ensembles for a given channel parameter instead of a given rate. The results however are easily extendable to the case of fixed rate ensembles. We call a sequence of degree distributions capacity achieving over a BIOSM channel, if the rate of the ensembles within the sequence can be made arbitrarily close to the channel capacity while maintaining the reliable communication. The design of provably capacity achieving sequences over general BIOSM channels is still an open problem. Another important problem of interest in LDPC codes is to design rate-compatible LDPC code schemes. In such a scheme, starting from a given primary ensemble called the parent code, we are interested in obtaining a set of codes with higher transmission rates, which can provide reliable transmission when the channel condition improves, by puncturing the parent code. For rate-compatibility, the design should be such that for two consecutive rates, the code with the higher rate can be constructed by puncturing the code with the lower rate. Starting from a parent code with performance close to capacity, the important challenge in a rate-compatible design is to also keep the performance of the punctured codes close to the capacity. More specifically, if the parent code is chosen from a capacity achieving sequence, all punctured codes have to be capacity achieving as average check node degree increases. To formulate the problem mathematically, imagine a parent code with rate R n from a capacity achieving sequence which can Forty-Seventh Annual Allerton Conference Allerton House, UIUC, Illinois, USA September 30 - October 2, 2009 978-1-4244-5871-4/09/$26.00 ©2009 IEEE 400 Authorized licensed use limited to: University of Massachusetts Amherst. Downloaded on July 09,2010 at 17:20:07 UTC from IEEE Xplore. Restrictions apply.