Pseudo-Binomial Degree Distributions from Derivative Matching Enrico Paolini and Marco Chiani DEIS / WiLAB, University of Bologna via Venezia 52, 47023 Cesena (FC), Italy {e.paolini,marco.chiani}@unibo.it Marc Fossorier ETIS ENSEA / UCP / CNRS UMR-8051 6, avenue du Ponceau, 95014, Cergy Pontoise, France mfossorier@ieee.org Abstract— In this paper, a method to design check- concentrated LDPC degree distributions for the erasure channel is proposed. This method is obtained taking a derivative matching approach. It consists of matching the first and high-order deriv- atives of the variable node decoder EXIT function and inverse check node decoder EXIT function in order to reduce the gap be- tween the two curves in the EXIT chart. A sufficient condition for a check-concentrated distribution to achieve derivative matching up to some order is first developed, and then a design algorithm is proposed exploiting this sufficient condition. A comparison with other deterministic design techniques is provided, revealing the potentialities of the proposed algorithm. I. I NTRODUCTION Currently, the most popular and effective tool for numerical optimization of low-density parity-check (LDPC) code degree distributions is represented by differential evolution (DE) [1], first applied to LDPC codes in [2]. The basic idea is to consider a population of N p degree distributions, all satisfying common constraints such as code rate and active variable node (VN) and check node (CN) degrees. At each step of the algorithm, a new population is generated from the old one through a recombination operation, in such a way that the maximum among the decoding thresholds is improved. Among the advantages of DE we can include its fastness (provided that the number of active degrees is not too large), the possibility of using it with different analysis tools, like density evolution [3] or Extrinsic Information Transfer (EXIT) chart [4], its general purpose nature which allows to apply it both to different communication channels and to code structures more general than LDPC codes [5], [6], and the possibility to perform constrained optimizations. Examples of constraints are represented by maximum number of active degrees, bounds on key parameters (such as λ ′ (0)ρ ′ (1) for LDPC codes), etc. Some drawbacks of DE are a general lack of control over the degree distribution optimization process, its pseudo-randomness which may lead to different final distrib- utions in correspondence with the same initial population, a certain dependence of the returned distribution on the starting population and, more importantly, an optimization time which may become impractical if the number of active degrees becomes too large. It is worthwhile noting that the algorithm pseudo-randomness and its dependence on the starting popu- lation are usually not an issue for LDPC degree distributions over the binary erasure channel (BEC). 0 100 200 300 400 500 600 700 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 L q * , 1 - R dc =6 dc = 10 Fig. 1. Binomial degree distribution chart (dc =6,..., 10). For each value of dc the upper curve is 1 - R and the lower curve is q * . For LDPC codes over the BEC solutions different from DE are available for obtaining degree distributions with a good threshold, namely, analytical solutions. The state of the art in this field is represented by [7]–[10]. In [7] and [8] the binomial sequence and the Tornado sequence were introduced, respec- tively, both able to achieve the BEC capacity. Slightly modified versions of these two sequences were successively regarded in [9] as special instances of a much more general technique for obtaining capacity achieving sequences for the BEC. A deterministic algorithm for degree distributions optimization is provided in [10] (see also Section IV-C). The binomial degree distribution introduced in [7] is check- regular and characterized by an edge-oriented VN distribution λ(x)= L  i=2 α ( α i−1 ) (−1) i α − L ( α L ) (−1) L+1 x i−1 , (1) where L denotes the maximum VN degree, d c the CN degree, α  (d c − 1) −1 and ( α N ) = α(α − 1) ... (α − N + 1)/(N !). A chart illustrating the behaviors of 1 − R (R being the code rate) and the threshold q ∗ as functions of L and d c is depicted in Fig. 1, from which it is evident how the q ∗ curve tightly matches the 1 − R curve for all the considered values of d c .A few additional observations are provided next. For each rate R, the larger d c the closer q ∗ to 1 − R (capacity achieving 2008 5th International Symposium on Turbo Codes and Related Topics 978-1-4244-2863-2/08/$25.00 ' 2008 IEEE 168 Authorized licensed use limited to: Universita degli Studi di Bologna. Downloaded on April 15, 2009 at 08:19 from IEEE Xplore. Restrictions apply.