Constraint Complexity of Realizations of Linear Codes on Arbitrary Graphs

ABSTRACT. A graphical realization of a linear code C consists of an assignment of the coordinates of C to the vertices of a graph, along with a specification of linear state spaces and linear "local constraint" codes to be associated with the edges and vertices, respectively, of the graph. The -complexity of a graphical realization is defined to be the largest dimension of any of its local constraint codes. -complexity is a reasonable measure of the computational complexity of a sumproduct decoding algorithm specified by a graphical realization. The main focus of this paper is on the following problem: given a linear code C and a graph G, how small can the -complexity of a realization of C on G be? As useful tools for attacking this problem, we introduce the Vertex-Cut Bound, and the notion of "vc-treewidth" for a graph, which is closely related to the well-known graphtheoretic notion of treewidth. Using these tools, we derive tight lower bounds on the -complexity o...

• Type: Journal
• Year: 2008
• Where: CORR
• Authors: Navin Kashyap
• Keywords: CORR 2008, Education, Graphical Realization, Linear Code, Local Constraint