2272 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 8, AUGUST 2009 Transactions Papers Evaluation and Design of Irregular LDPC Codes Using ACE Spectrum Dejan Vukobratovi´ c, Student Member, IEEE, and Vojin Šenk, Member, IEEE Abstract—The construction of finite-length irregular LDPC codes with low error floors is currently an attractive research problem. In particular, for the binary erasure channel (BEC), the problem is to find the elements of selected irregular LDPC code ensembles with the size of their minimum stopping set being maximized. Due to the lack of analytical solutions to this problem, a simple but powerful heuristic design algorithm, the approximate cycle extrinsic message degree (ACE) constrained design algorithm, has recently been proposed. Building upon the ACE metric associated with a cycle in a code graph, we introduce the ACE spectrum of LDPC codes as a useful tool for evaluation of codes from selected irregular LDPC code ensembles. Using the ACE spectrum, we generalize the ACE constrained design algorithm, making it more flexible and efficient. We justify the ACE spectrum approach through examples and simulation results. Index Terms—Approximate cycle extrinsic message degree (ACE), error floor, irregular low-density parity-check (LDPC) codes, iterative decoding, stopping sets. I. I NTRODUCTION T HE construction of short and medium length irregular LDPC codes with low error floors is currently an attrac- tive research area. Many practical systems are constrained by delay, memory, processing or similar requirements that limit the maximum desirable code length. Asymptotic behavior of LDPC code ensembles predicted by density evolution (DE) [1] analysis should not be the only design criterion, as it is known that short-length LDPC codes designed using only DE guidelines behave poorly. Instead, one has to analyze finite-length behavior of the iterative belief-propagation (BP) decoding algorithm to design LDPC codes that perform well. Usually, the task is to design LDPC code graphs free of the subgraphs that are proved to introduce problems for the iterative BP decoder. LDPC code design and simulation results presented in this paper assume transmission over the AWGN (additive white Paper approved by O. Milenkovic, the Editor for Coding Theory and Applications of the IEEE Communications Society. Manuscript received October 21, 2007; revised July 15, 2008. The authors are with Dept. of Power, Electronics and Communication Engineering, University of Novi Sad, Novi Sad, Serbia (e-mail: {dejanv, ram_senk}@uns.ns.ac.yu.). The material in this paper was presented in part at the IEEE International Conference on Communications ICC 2007, Glasgow, UK, 24-28. June 2007. This work was supported by Grant for research in technology No. TR- 11022, awarded by the Republic Ministry of Science, Republic of Serbia. Digital Object Identifier 10.1109/TCOMM.2009.08.070548 Gaussian noise) channel. A significant amount of work has been done recently in the area of finite-length behavior of the iterative decoding of LDPC codes over the AWGN channel [2]. For simplicity, however, we use the results that address finite-length behavior of the iterative decoding over the BEC for our code design. Designing good LDPC codes for the BEC produces good LDPC codes for the AWGN channel as well, making this design approach attractive [3]. Finite-length behavior of LDPC codes, decoded iteratively over the BEC, is determined by the subgraphs of an LDPC code graph called stopping sets [4]. Small stopping sets deteriorate the performance of LDPC codes in the error-floor region. The design method that optimizes irregular LDPC codes with respect to their minimum stopping set size (i.e, their stopping distance [5]) is yet unknown, which is one of the most challenging open problems in the LDPC code design (see [6] and references therein). However, the recently proposed ACE constrained design algorithm [7] heuristically produces LDPC codes with improved error-floor performance by influencing stopping sets via conditioning cycles. This paper builds upon the ACE constrained LDPC code design. We introduce the ACE spectrum as an efficient tool for quick evaluation of finite-length irregular LDPC codes. The goal is to identify a subset of the selected irregular LDPC code ensemble containing the candidates with excellent error- correcting performance, as well as to develop code design algorithms that output the LDPC codes from this subset. Classifying LDPC codes with respect to their ACE spectra, we identify the LDPC codes with extremal ACE spectrum properties as the subset of our interest. We propose the gen- eralized ACE constrained design algorithm to design LDPC codes from this subset. The paper is organized as follows. Following the intro- ductory section, Section II briefly reviews the ACE metric associated with a cycle in the LDPC code graph and the idea behind the ACE constrained design, proposed in [7]. Based on the ACE metric, Section III introduces the ACE spectrum of LDPC codes, which can be calculated efficiently for any LDPC code. In Section IV, we propose a generalized version of the ACE constrained LDPC code design based on the ACE spectrum. Using examples and simulation results, we show that the generalized design is more flexible and efficient than the original design. The paper is concluded in Section V. 0090-6778/09$25.00 c  2009 IEEE