University of Illinois, Champaign-Urbana Alexander Vardy RIA: Channel Codes for Digital Communications and Storage Systems Reliable transmission and storage of information requires the use of error correcting channel codes to protect the data against errors introduced by the channel noise or imperfections of the recording medium. Error correcting codes are implemented in most of the modern communications and storage systems ranging from household compact-disc players through telephone line modems, computer memories, and mobile radio networks to satellite and deep space communications. In this project we investigate two general types of error correcting codes, known as block and lattice codes, using a novel dynamical approach. Since a block code is essentially subset of finite field, while a lattice is a discrete collection of dimensional real points with certain prescribed distance properties, they have been conventionally treated as geometric or algebraic entities. However, as is just now being realized, block and lattice codes may as well be regarded as dynamical systems. The latter approach has several profound advantages over the conventional practice. One of the objectives is to exploit these advantages in an attempt to find new codes, better than presently known. Another objective is to provide bounds on the decoding complexity and develop more efficient maximum-likelihood decoders, which substantially advance the current performance achievable for a given decoder complexity. Furthermore, we study the precise trade-off between complexity and performance in block and lattice error correcting codes, from both theoretical and practical standpoints. Progress along these lines would enable the designer of a communication system to obtain larger coding gains for the same bandwidth, power, and complexity constraints. Also treated are modulation codes for input constrained channels used to encode information into a particular set of sequences admitted by the channel. These codes have widespread use in a variety of information storage applications, such as magnetic or magneto-optic recording systems. New multi-dimensional modulation codes are currently being developed for the emerging technology of holographic storage. Most of the modulation codes in use today are constructed using tools from symbolic dynamics. Taking the point of view of block codes as dynamical systems makes it natural to consider applying results from algebraic coding theory for the design of modulation codes. We will use this approach to develop more efficient encoders for high order spectral null codes and multidimensional modulation codes for holographic recording. The possibility of integrating a prescribed error correcting capability within such modulation encoders will also be studied.
