On the Design of Universal LDPC Codes

📝 Abstract

Low-density parity-check (LDPC) coding for a multitude of equal-capacity channels is studied. First, based on numerous observations, a conjecture is stated that when the belief propagation decoder converges on a set of equal-capacity channels, it would also converge on any convex combination of those channels. Then, it is proved that when the stability condition is satisfied for a number of channels, it is also satisfied for any channel in their convex hull. For the purpose of code design, a method is proposed which can decompose every symmetric channel with capacity C into a set of identical-capacity basis channels. We expect codes that work on the basis channels to be suitable for any channel with capacity C. Such codes are found and in comparison with existing LDPC codes that are designed for specific channels, our codes obtain considerable coding gains when used across a multitude of channels.

💡 Analysis

Low-density parity-check (LDPC) coding for a multitude of equal-capacity channels is studied. First, based on numerous observations, a conjecture is stated that when the belief propagation decoder converges on a set of equal-capacity channels, it would also converge on any convex combination of those channels. Then, it is proved that when the stability condition is satisfied for a number of channels, it is also satisfied for any channel in their convex hull. For the purpose of code design, a method is proposed which can decompose every symmetric channel with capacity C into a set of identical-capacity basis channels. We expect codes that work on the basis channels to be suitable for any channel with capacity C. Such codes are found and in comparison with existing LDPC codes that are designed for specific channels, our codes obtain considerable coding gains when used across a multitude of channels.

📄 Content

arXiv:0806.0036v1 [cs.IT] 30 May 2008 On the Design of Universal LDPC Codes Ali Sanaei, Mahdi Ramezani, and Masoud Ardakani Department of Electrical and Computer Engineering, University of Alberta, Canada Email: {sanaei,ramezani,ardakani}@ece.ualberta.ca Abstract—Low-density parity-check (LDPC) coding for a mul- titude of equal-capacity channels is studied. First, based on numerous observations, a conjecture is stated that when the belief propagation decoder converges on a set of equal-capacity channels, it would also converge on any convex combination of those channels. Then, it is proved that when the stability condition is satisfied for a number of channels, it is also satisfied for any channel in their convex hull. For the purpose of code design, a method is proposed which can decompose every symmetric channel with capacity C into a set of identical-capacity basis channels. We expect codes that work on the basis channels to be suitable for any channel with capacity C. Such codes are found and in comparison with existing LDPC codes that are designed for specific channels, our codes obtain considerable coding gains when used across a multitude of channels. I. INTRODUCTION Design of codes for specific channels is a mature subject. A code optimized for one channel, however, may not perform well if used on other channel types [1]. Thus, recently, there has been an increasing interest in universal codes, i.e., codes that perform well on a multitude of channels [2]–[4]. Such codes reduce system complexity by removing a need for frequent code changes in the system and by allowing for once- and-for-all coding solutions. Low-density parity-check (LDPC) codes [5] are extremely strong error correcting codes. Interestingly, various authors have observed “universal properties” of these codes [3], [4], [6]–[8]. Chung [6] points out that LDPC codes optimized for the Gaussian channel perform well on some other channels such as the Rayleigh channel. In a more general setup, Shi and Wesel [4] discuss the universal properties of finite block length codes. Peng et al. [8] design LDPC codes for a number of channels (in this case, the Gaussian channel, the binary erasure channel (BEC) and the Rayleigh channel). They argue that for a set of channels, usually one channel can be taken as the surrogate. They design the code only for the surrogate channel. This code works satisfactorily on all the given channels, but not necessarily on other channel types. Despite some universal properties of LDPC codes, the performance of a code designed for one channel, can be poor on another channel with the same capacity. For example, a rate one-half LDPC code with maximum node degree of 100 (taken from [9]), achieving more than 99.7% of the capacity of a BEC with capacity C = 0.5, does not converge on a binary symmetric channel (BSC) with capacity C = 0.63. In other words, this code does not achieve even 80% of the capacity of the BSC. In this work, we design LDPC codes that have strong universal properties. For a given channel capacity C, we find LDPC codes that perform well on any channel with this capacity. The main body of this paper consists of: (1) studying code design and stability analysis for convex combination of N channels of equal capacity, and (2) decomposition of all channels with capacity C into a number of basis channels of the same capacity. In Section III, we study code design for the convex com- binations of a set of equal-capacity channels (see Section III for definition). We conjecture that under belief propagation decoding, to design a code for all the convex combinations of N binary-input symmetric-output (BISO) channels, it is sufficient to find a code which converges only on those N channels. We also prove that when stability condition [10] is satisfied for N BISO channels, it is also satisfied for all their convex combinations. As a result, if a set of basis channels with capacity C can be found, a code designed only for the basis channels has strong universal properties. In Section IV, a channel decomposition method is suggested which decomposes any BISO channel with capacity C in terms of basis channels of the same capacity with nonnegative coefficients. This decomposition method is not similar to existing techniques that decompose the channel over BSCs of various capacities [11]. Our technique is exhaustive, i.e., any channel with a given capacity can be spanned with nonnegative coefficients over our suggested basis. For any given capacity C, a code designed only for our suggested basis channels is expected to have strong universal behavior over all channels with capacity C. Examples are provided in Section V. Simulations confirm that compared to codes designed for specific channels, our codes have signifi- cantly better universal properties. II. PRELIMINARIES: LDPC CODE DESIGN AND SYMMETRIC CHANNEL REPRESENTATIONS In this paper, an ensemble of LDPC codes is defined by a pair of distributions (λ, ρ) in the polynomial form, i.e.,

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