# Codes on graphs: normal realizations

@article{Forney2000CodesOG, title={Codes on graphs: normal realizations}, author={G. David Forney}, journal={2000 IEEE International Symposium on Information Theory (Cat. No.00CH37060)}, year={2000}, pages={9-} }

Wiberg et al. (see European Transactions on Telelecommunications, vol.6, p.513-25, Sept./Oct. 1995) proposed graphical code realizations using three kinds of elements: symbol variables, state variables and local constraints. We focus on normal realizations, namely Wiberg-type realizations in which all symbol variables have degree 1 and state variables have degree 2. A natural graphical model of a normal realization represents states by leaf edges, states by ordinary edges, and local constraints…

## 636 Citations

### Codes on graphs: constraint complexity of cycle-free realizations of linear codes

- Computer ScienceIEEE Transactions on Information Theory
- 2003

Cycle-free graphical realizations of linear codes generalize trellis realizations and can yield reductions in decoding complexity even when they do not reduce constraint complexity.

### Constraint Complexity of Realizations of Linear Codes on Arbitrary Graphs

- Computer Science, MathematicsIEEE Transactions on Information Theory
- 2009

The vertex-cut bound, and the notion of "vc-treewidth" for a graph, which is closely related to the well-known graph-theoretic notion of treewidth, are introduced and derived to derive tight lower bounds on the kappa-complexity of any realization of C on G, which enable us to conclude that good error-correcting codes can have low- complexity realizations only on graphs with large vc-Treewidth.

### The Tradeoff Between Cyclic Topology and Complexity in Graphical Models of Linear Codes

- Computer Science
- 2006

A new bound is introduced the tree-inducing cut-set bound which can be viewed as a generalization of the square-root bound to graphical models with arbitrary cyclic topologies.

### Codes on Graphs : Observability , Controllability , and Local

- Mathematics
- 2012

This paper investigates properties of realizations of linear or group codes on general graphs that lead to local reducibility. Trimness and properness are dual properties of constraint codes. A…

### Syndrome realizations of linear codes and systems

- Computer ScienceProceedings 2003 IEEE Information Theory Workshop (Cat. No.03EX674)
- 2003

It is shown that syndrome realizations are essentially the same as the restricted state realizations of Forney, G.D., Jr. (see IEEE Trans. Inf. Theory, vol.47, p.520-48, 2001), which, in some respects, appear to be more fundamental.

### Codes on Graphs: Observability, Controllability, and Local Reducibility

- MathematicsIEEE Transactions on Information Theory
- 2013

This paper investigates properties of realizations of linear or group codes on general graphs that lead to local reducibility and finds that trimness and properness are dual properties of constraint codes that are locally reducible.

### Iterative Decoding of Codes on Graphs

- Computer Science
- 2006

This report furthers the understanding of the failures in iterative decoders for the binary symmetric channel, and proposes an algorithm that aims to alleviate this drawback by constructing an equivalent graph representation that is free of four cycles.

### Conditionally Cycle-Free Graphical Models for Coset Codes

- Computer ScienceIEEE Transactions on Information Theory
- 2012

Examples of a number of families of codes - including first-order Reed-Muller (RM) and the Delsarte-Goethals codes - are provided for which the proposed procedure yields optimal soft-in soft-out (SISO) decoding algorithms that are less complex than the best known trellis-based algorithms.

### The Complexity Limits of Graphical Models for Linear Codes

- Computer Science2007 IEEE Information Theory Workshop
- 2007

The tree-inducing cut-set bound (TI-CSB) provides a characterization of the tradeoff between complexity and density in the family of graphical models for a given code, and can be used to show that an r th-root complexity reduction requires the introduction of at least r(r - 1)/2 cycles thus generalizing the square-root bound to graphical models with arbitrary cyclic topologies.

### Convolutional Codes on Trees and Cayley Graphs

- Computer Science, Mathematics
- 2001

This work generalizes the notion of discrete-time index sets to in nite regular trees and introduces a generalization of conventional tail-biting to deal with the eÆcient termination of convolutional codes on trees.

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