Applied Algebra, Algebraic Algorithms and Error-Correcting Codes

There is intense current interest in the subject of “codes on graphs.” Graphical models for codes arise from state realizations. Normal realizations may be assumed without loss of generality. A normal realization has a dual which necessarily generates the dual code. Conventional trellises and tail-biting realizations are inherently normal. Efficient realizations of Reed-Muller codes are presented. Low-density parity-check codes use generic parity-check realizations. Algebraic tools may improve such graph-based codes.