On Quadratic Inverses for Quadratic Permutation Polynomials over Integer Rings

An interleaver is a critical component for the channel coding performance of turbo codes. Algebraic constructions are of particular interest because they admit analytical designs and simple, practical hardware implementation. Sun and Takeshita have recently shown that the class of quadratic permutation polynomials over integer rings provides excellent performance for turbo codes. In this correspondence, a necessary and sufficient condition is proven for the existence of a quadratic inverse polynomial for a quadratic permutation polynomial over an integer ring. Further, a simple construction is given for the quadratic inverse. All but one of the quadratic interleavers proposed earlier by Sun and Takeshita are found to admit a quadratic inverse, although none were explicitly designed to do so. An explanation is argued for the observation that restriction to a quadratic inverse polynomial does not narrow the pool of good quadratic interleavers for turbo codes.Comment: Submitted as a Correspondence to the IEEE Transactions on Information Theory Submitted : April 1, 2005 Revised : Nov. 15, 200

An Analysis of the Connections Between Layers of Deep Neural Networks

We present an analysis of different techniques for selecting the connection be- tween layers of deep neural networks. Traditional deep neural networks use ran- dom connection tables between layers to keep the number of connections small and tune to different image features. This kind of connection performs adequately in supervised deep networks because their values are refined during the training. On the other hand, in unsupervised learning, one cannot rely on back-propagation techniques to learn the connections between layers. In this work, we tested four different techniques for connecting the first layer of the network to the second layer on the CIFAR and SVHN datasets and showed that the accuracy can be im- proved up to 3% depending on the technique used. We also showed that learning the connections based on the co-occurrences of the features does not confer an advantage over a random connection table in small networks. This work is helpful to improve the efficiency of connections between the layers of unsupervised deep neural networks