The Complexity Limits of Graphical Models for Linear Codes

Since the discovery of turbo codes in the 1990s, much of the work concerning graphical models for codes has focused on constructing models in order to define a new code. This work focuses on the theoretical foundations of the inverse problem to code construction -extracting a graphical model for a given (fixed) code. Specifically, limits on the space of graphical models for a given code are examined. A number of results concerning these limits (e.g., the cut-set bound, the square-root bound) are first reviewed. Following this review, recent results concerning these limits are presented. It is first shown that the Tanner graphs of many classical block codes necessarily contain many short cycles. It is then shown that cycle-free binary graphical models do not support good codes thus generalizing Trachtenberg's work concerning the existence of cycle-free Tanner graphs. Finally, and most significantly, the tree-inducing cut-set bound (TI-CSB) is derived and discussed. The TI-CSB provides a characterization of the tradeoff between complexity and density in the family of graphical models for a given code. For example, the TI-CSB can be used to show that an
r

^th
-root complexity reduction (with respect to the least-complex cycle-free model) requires the introduction of at least
r
(
r
- 1)/2 cycles thus generalizing the square-root bound to graphical models with arbitrary cyclic topologies.