Weight distribution

Abstract: Gleason has described the general form that the weight distribution of a self-dual code over GF (2) and GF (3) can have. We give an explicit formula for this weight distribution when the minimum distance d between codewords is made as large as possible. It follows that for self-dual codes of length n over GF (2) with all weights divisible by 4, d ⩽ 4[ n /24] + 4; and for self-dual codes over GF (3), d ⩽ 3[ n /12] + 3; where the square brackets denote the integer part. These results improve on the Elias bound. A table of this extremal weight distribution is given in the binary case for n ⩽ 200 and n = 256.

Abstract: MacKay's (1992) Bayesian framework for backpropagation is a practical and powerful means to improve the generalization ability of neural networks. It is based on a Gaussian approximation to the posterior weight distribution. The framework is extended, reviewed, and demonstrated in a pedagogical way. The notation is simplified using the ordinary weight decay parameter, and a detailed and explicit procedure for adjusting several weight decay parameters is given. Bayesian backprop is applied in the prediction of fat content in minced meat from near infrared spectra. It outperforms "early stopping" as well as quadratic regression. The evidence of a committee of differently trained networks is computed, and the corresponding improved generalization is verified. The error bars on the predictions of the fat content are computed. There are three contributors: The random noise, the uncertainty in the weights, and the deviation among the committee members. The Bayesian framework is compared to Moody's GPE (1992). Finally, MacKay and Neal's automatic relevance determination, in which the weight decay parameters depend on the input number, is applied to the data with improved results.

Abstract: We derive the average weight distribution function and its asymptotic growth rate for low-density parity-check (LDPC) code ensembles. We show that the growth rate of the minimum distance of LDPC codes depends only on the degree distribution pair. It turns out that capacity-achieving sequences of standard (unstructured) LDPC codes under iterative decoding over the binary erasure channel (BEC) known to date have sublinearly growing minimum distance in the block length

Abstract: With any fixed prime number p and positive integer N , not divisible by p , there is associated an infinite sequence of cyclic codes. In a previous article it was shown that a theorem of Davenport-Hasse reduces the calculation of the weight distributions for this whole sequence of codes to a single calculation (essentially that of calculating the weight distribution for the simplest code of the sequence). The primary object of this paper is the development of machinery which simplifies this remaining calculation. Detailed examples are given. In addition, tables are presented which essentially solve the weight distribution problem for all such binary codes with N