Capacity-Achieving Ensembles of Accumulate-Repeat-Accumulate Codes for the Erasure Channel with Bounded Complexity

The paper introduces ensembles of accumulate-repeat-accumulate (ARA) codes which asymptotically achieve capacity on the binary erasure channel (BEC) with {\em bounded complexity}, per information bit, of encoding and decoding. It also introduces symmetry properties which play a central role in the construction of capacity-achieving ensembles for the BEC with bounded complexity. The results here improve on the tradeoff between performance and complexity provided by previous constructions of capacity-achieving ensembles of codes defined on graphs. The superiority of ARA codes with moderate to large block length is exemplified by computer simulations which compare their performance with those of previously reported capacity-achieving ensembles of LDPC and IRA codes. The ARA codes also have the advantage of being systematic.Comment: Submitted to IEEE Trans. on Information Theory, December 1st, 2005. Includes 50 pages and 13 figure

Normal Factor Graphs and Holographic Transformations

This paper stands at the intersection of two distinct lines of research. One line is "holographic algorithms," a powerful approach introduced by Valiant for solving various counting problems in computer science; the other is "normal factor graphs," an elegant framework proposed by Forney for representing codes defined on graphs. We introduce the notion of holographic transformations for normal factor graphs, and establish a very general theorem, called the generalized Holant theorem, which relates a normal factor graph to its holographic transformation. We show that the generalized Holant theorem on the one hand underlies the principle of holographic algorithms, and on the other hand reduces to a general duality theorem for normal factor graphs, a special case of which was first proved by Forney. In the course of our development, we formalize a new semantics for normal factor graphs, which highlights various linear algebraic properties that potentially enable the use of normal factor graphs as a linear algebraic tool.Comment: To appear IEEE Trans. Inform. Theor