Statistical physics of regular low-density parity-check error-correcting codes

Abstract

A variation of Gallager error-correcting codes is investigated using statistical mechanics. In codes of this type, a given message is encoded into a codeword that comprises Boolean sums of message bits selected by two randomly constructed sparse matrices. The similarity of these codes to Ising spin systems with random interaction makes it possible to assess their typical performance by analytical methods developed in the study of disordered systems. The typical case solutions obtained via the replica method are consistent with those obtained in simulations using belief propagation decoding. We discuss the practical implications of the results obtained and suggest a computationally efficient construction for one of the more practical configurations.

Finitely connected vector spin systems with random matrix interactions

“…The physical transition 10 away from the paramagnetic state, as the connectivity is increased from c = 0 for fixed βJ , is the one with the largest value of c −1…”

“…In particular due to the unexpectedly rich and varied range of multi-disciplinary applications of finite connectivity replica techniques which emerged subsequently in, for example, spin-glass modelling [6][7][8][9], error correcting codes [10][11][12][13], theoretical computer science [14][15][16][17], recurrent neural networks [18][19][20] and 'small-world' networks [21], this field is presently enjoying a renewed interest and popularity. Until very recently, analysis was limited to the equilibrium properties of such models, but now attention has also turned to the dynamics of finitely connected spin systems [22][23][24][25], using combinatorial and generating functional methods.…”

Parallel dynamics of disordered Ising spin systems on finitely connected random graphs

“…Furthermore, finitely connected models are found to exhibit many interesting and complex new features which are worthy of analysis. Their equilibrium properties have been studied in the context of spin glasses [1,2,3,4,5,6], error correcting codes [7,8,9], satisfiability problems [10,11,12,13], neural networks [14,15] and 'small world' models [16,17]. Such analyses involve order parameter functions, which generalize the replica matrices of [18].…”

A mean field theory of coded CDMA systems

“…As shown in [17], when the row weight is not so large, the regular LDPC-coded CDMA system has two kinds of phase transitions, the decoding threshold and the information-theoretic threshold, just as those found in statistical mechanical analysis of single-user LDPC codes [8,15,16]. The former is also called the dynamical transition point, and the only stable solution is the perfect ferromagnetic solution when noise level is smaller than that, while the suboptimal ferromagnetic solution appears when noise level exceeds the point.…”

“…The third term of the right-hand side of (18) and the order parameters (23), for the regular LDPC-coded CDMA systems, can be evaluated on the basis of the calculation given in [16]: We therefore omit details of the derivation. Denoting the third term of (18) divided by the codelength M as h z , and letting L and C be the row and column weights, respectively, we obtain, in the infinite codelength limit (M → ∞ with R fixed),…”