Coding Theory
Our research explores the relations among coding theory, information theory, and related areas of computer science and mathematics. In error control coding, our team is concerned with the design, performance analysis and decoding of efficient modern error control systems such as wavelet codes, rateless codes, low density parity check codes (LDPC) and other types of iterative coding schemes. We are particularly interested in coding techniques for erasure channels, unequal error protecting codes, rate compatible codes, nonuniform codes, and two-dimensional codes for digital communication/storage systems and wireless ad hoc networks.
In the past, the developments in wavelet decompositions and filter bank theory focused almost exclusively on real or complex fields. We develop the theory of finite-field wavelet transforms that provides a general wavelet decomposition of sequences defined over finite fields. This is an approach that has a rich history in signal processing for the representation of real-valued signals, but it has been lacking in the finite-field case. We introduce elementary paraunitary building blocks and a factorization technique that are specialized to obtain a complete realization for all paraunitary filter banks over fields of characteristic two. The work has led to a basis for a unified development of finite field wavelets and error control coding. We derived wavelet representations of many previously known codes. We find new codes such as self-dual codes and two-dimensional wavelet codes. We develop maximum distance wavelet codes that are the best possible codes in that they correct the maximum number of errors for a given amount of redundancy. We introduce new time-varying convolutional codes that can be decoded faster than conventional convolutional codes. random coding approach to jointly solve for packet authentication and availability.
