An iterative algorithm for the multiuser detection problem that arises in code division multiple access (CDMA) systems is developed on the basis of Pearl's belief propagation (BP). We show that the BP-based algorithm exhibits nearly optimal performance in a practical time scale by utilizing the central limit theorem and self-averaging property appropriately, whereas direct application of BP to the detection problem is computationally difficult and far from practical. We further present close relationships of the proposed algorithm to the Thouless-Anderson-Palmer approach and replica analysis known in spin-glass research.

Abstract. We consider the problem of reconstructing an N -dimensional continuous vector x from P constraints which are generated by its linear transformation under the assumption that the number of non-zero elements of x is typically limited to ρN (0 ≤ ρ ≤ 1). Problems of this type can be solved by minimizing a cost function with respect to the L p -norm ||x|| p = lim ǫ→+0 N i=1 |x i | p+ǫ , subject to the constraints under an appropriate condition. For several p, we assess a typical case limit α c (ρ), which represents a critical relation between α = P/N and ρ for successfully reconstructing the original vector by minimization for typical situations in the limit N, P → ∞ with keeping α finite, utilizing the replica method. For p = 1, α c (ρ) is considerably smaller than its worst case counterpart, which has been rigorously derived by existing literature of information theory.

Abstract. -An approach to analyze the performance of the code division multiple access (CDMA) scheme, which is a core technology used in modern wireless communication systems, is provided. The approach characterizes the objective system by the eigenvalue spectrum of a cross-correlation matrix composed of signature sequences used in CDMA communication, which enable us to handle a wider class of CDMA systems beyond the basic model reported by Tanaka in Europhys. Lett., 54 (2001) 540. The utility of the scheme is shown by analyzing a system in which the generation of signature sequences is designed for enhancing the orthogonality.Introduction. -Over the last decade, the scope of statistical mechanics has rapidly expanded beyond its original goal of analyzing many-body problems that arise when dealing with material objects. Information theory is a major source of problems, and research activity aimed at solving these problems is becoming popular. Identifying information bits with Ising spins, many problems in information theory, such as error correcting/compression codes [1-10] and cryptosystems [11,12], can be formulated as virtual many-body systems that are subject to disordered interactions. Analysis of the formulated problems using techniques of statistical mechanics has provided various nontrivial results that have not been obtained by conventional methods of information theory [13,14].Code division multiple access (CDMA), which is a core technology used in modern wireless communication, is an example of the successful application of such a statistical mechanical approach. This technology realizes simultaneous communication between multiple users and a single base station by modulating each user's bit signal (symbol) into a sequence of random pattern, termed the signature sequence [15]. CDMA has already been employed in thirdgeneration mobile phone systems and wireless LANs.Tanaka (2001) showed that the replica method of statistical mechanics enables the accurate assessment of the communication performance of a basic CDMA model in which users' sequences are generated independently of each other in a large system limit [16,17]. This research

PACS. 89.90+n -Other areas of general interest to physicists. PACS. 89.70+c -Information science. PACS. 05.50+q -Lattice theory and statistics; Ising problems.Abstract. -We investigate the performance of error-correcting codes, where the code word comprises products of K bits selected from the original message and decoding is carried out utilizing a connectivity tensor with C connections per index. Shannon's bound for the channel capacity is recovered for large K and zero temperature when the code rate K/C is finite. Close to optimal error-correcting capability is obtained for finite K and C. We examine the finite-temperature case to assess the use of simulated annealing for decoding and extend the analysis to accommodate other types of noisy channels.Error-correcting codes are of significant practical importance as they provide mechanisms for retrieving the original message after possible corruption due to noise during transmission. They are being used extensively in most means of information transmission from satellite communication to the storage of information on hardware devices. The coding efficiency, measured in the percentage of informative transmitted bits, plays a crucial role in determining the speed of communication channels and the effective storage space on hard-disks. Rigorous bounds [1] have been derived for the maximal channel capacity for which codes, capable of achieving arbitrarily small error probability, can be found. However, existing practical errorcorrecting codes do not saturate this bound and the quest for more efficient error-correcting codes has been going on ever since.A new family of error-correcting codes, based on insights gained from the statistical mechanical analysis of Ising spin models, has recently been suggested by Sourlas [2], investigating the use of statistical mechanics for constructing and investigating novel coding methods [3,12]. However, the codes suggested and analyzed so far are of no practical significance as they imply an infinite ratio between the length of the transmitted word and the original message. Consequently, they had little impact on the design and the understanding of practical codes.

PACS. 89.70+c -Information science. PACS. 89.90+n -Other areas of general interest to physicists. PACS. 02.50−r -Probability theory, stochastic processes, and statistics.Abstract. -We employ two different methods, based on belief propagation and TAP, for decoding corrupted messages encoded by employing Sourlas's method, where the code word comprises products of K bits selected randomly from the original message. We show that the equations obtained by the two approaches are similar and provide the same solution as the one obtained by the replica approach in some cases (K = 2). However, we also show that for K ≥ 3 and unbiased messages the iterative solution is sensitive to the initial conditions and is likely to provide erroneous solutions; and that it is generally beneficial to use Nishimori's temperature, especially in the case of biased messages.Belief networks [1], also termed Bayesian networks, and influence diagrams are diagrammatic representations of joint probability distributions over a set of variables. The set of variables is usually represented by the vertices of a graph, while arcs between vertices represent probabilistic dependences between variables. Belief propagation provides a convenient mathematical tool for calculating iteratively joint probability distributions of variables, and have been used in a variety of cases to assess conditional probabilities and interdependences between variables in complex systems. One of the most recent uses of belief propagation is in the field of error-correcting codes, especially for decoding corrupted messages [2] (for a review of graphical models and their use in the context of error-correcting codes see [3]).Error-correcting codes provide a mechanism for retrieving the original message after corruption due to noise during transmission. A new family of error-correcting codes, based on insights gained from statistical mechanics, has recently been suggested by Sourlas [4]. These codes can be mapped onto the many-body Ising spin problem and can thus be analysed using methods adopted from statistical physics [5][6][7][8][9].In this letter we will examine the similarities and differences between the belief propagation (BP) and TAP approaches, used as decoders in the context of error-correcting codes. We will then employ these approaches to examine a few specific cases and compare the results to the solutions obtained using the replica method [8]. This will enable us to draw some conclusions on the efficacy of the TAP/BP approach in the context of error-correcting codes.