James W. Bond

Low Density Parity Check Codes Based on Sparse Matrices with No Small Cycles

In this paper we give a systematic construction of matrices with constant row weights and column weights and arbitrarily large girths. This resolves a problem raised by D. MacKay. The matrices are used in the generator matrices of linear codes. We give the experiment performance results for codes whose associated matrices have girth 8. We also give a randomized construction of matrices with constant row sums and column sums and few 4-cycles. The codes generated using the matrices are used to encode bit streams for a Gaussian channel and decoded using a decoding algorithm that combines features of the algorithms given by MacKay and Cheng and McEliece. The experimental performance results for codes generated using the random matrices are compared to those of the systematically constructed codes. The results show that the codes generated using the random codes with smaller block sizes perform as well as the systematic codes with bigger block sizes. The performance of the systematic codes, for specified weights, can be used to tailor the random codes. MATLAB routines for the construction for the girth 8 case and a special girth 4 case are included.

Linear-Congruence Constructions of Low-Density Parity-Check Codes

Low-Density Parity-Check codes (LDPCC) with Iterative Belief Propagation (Message Passing) decoding are attractive alternatives to Turbo codes. LDPCC previously discussed in the literature have involved matrices constructed using random techniques. In this paper, we discuss construction techniques for LDPCC involving multiple permutation matrices, each specified by a linear congruence. Construction options depend on the size of the parity-check matrix and the rate of the code. We relate desirable properties of the code to the parameters defining the linear congruences specifying the permutation matrices used to construct the code. For example, codes with few or no 4-cycles can be readily constructed. We summarize the construction options and describe selection processes for the parameters of the congruences. We then provide performance results for regular parity-check matrices constructed by random and the linear-congruence techniques for rate 1/2 transmit block-size 980 and rate 4/7 transmit block-size 847 codes. We introduce a symmetric channel model for decoding with the iterative belief propagation algorithm and describe its use as a heuristic for deciding whether a code is likely better or worse than most codes of the given rate and block size.

The Euclidean Algorithm and Primitive Polypomials over Finite Fields

In this paper, we study the quotients that arise when the Euclidean algorithm is applied to a primitive polynomial and xs - 1. We analyze the asymptotic behavior of the the number of terms of the quotients as n → ∞. This problem comes from the study of Low-Density Parity-Check codes. We obtain two characterizations of primitive polynomials over the field with two elements that are based on the number of nonzero terms in two polynomials obtained via division and the Euclidean algorithm with polynomials of the form xs - 1. The analogous results do not hold for general finite fields but do restrict the order of the polynomials to a small set of positive integers with specific forms.

Implicit models of Gaussian mixture densities and locally optimum detectors

Abstract In the classical kernel approximation of a probability density function, a fixed zero-mean smooth approximation of the Dirac delta function is translated to each sample value of the random variable to form a family of density functions. The sample values are the means of the density functions in the corresponding family. The kernel approximation is the average over all members of this family. In certain applications, there may be too few sample values to give a good approximation. In this paper, we propose an approach that utilizes the sample values as the variances of a family of density functions. This approach works particularly well with a Gaussian mixture density.