On Minimal Tree Realizations of Linear Codes

Abstract

ABSTRACT. A tree decomposition of the coordinates of a code is a mapping from the coordinate set to the set of vertices of a tree. A tree decomposition can be extended to a tree realization, i.e., a cycle-free realization of the code on the underlying tree, by specifying a state space at each edge of the tree, and a local constraint code at each vertex of the tree. The constraint complexity of a tree realization is the maximum dimension of any of its local constraint codes. A measure of the complexity of maximum-likelihood decoding for a code is its treewidth, which is the least constraint complexity of any of its tree realizations.It is known that among all tree realizations of a code that extends a given tree decomposition, there exists a unique minimal realization that minimizes the state space dimension at each vertex of the underlying tree. In this paper, we give two new constructions of these minimal realizations. As a by-product of the first construction, a generalization of the state-merging procedure for trellis realizations, we obtain the fact that the minimal tree realization also minimizes the local constraint code dimension at each vertex of the underlying tree. The second construction relies on certain code decomposition techniques that we develop. We further observe that the treewidth of a code is related to a measure of graph complexity, also called treewidth. We exploit this connection to resolve a conjecture of Forney's regarding the gap between the minimum trellis constraint complexity and the treewidth of a code. We present a family of codes for which this gap can be arbitrarily large.

Constraint Complexity of Realizations of Linear Codes on Arbitrary Graphs

“…The identity in (4), which may be viewed as a generalization of the statement of Theorem 4.6 in [14] for conventional trellis realizations, will not be proved here as it would be an unnecessary deviation from the main line of our development. It suffices to say that (4) follows from Theorem 3.4 in [11] by first verifying that it indeed holds for any minimal tree realization M(C; T, ω), and then observing that the difference between κ + (Γ) and σ + (Γ) is preserved by the state-merging process mentioned in the statement of that theorem. Finally, we remark that the state max-complexity of a graphical realization cannot exceed the constraint max-complexity of the realization.…”

“…Graphical Realizations of Codes. The development in this section is based on the exposition of Forney [6], [7]; see also [8], [11]. Let I be a finite index set.…”

“…An essential graphical model (G, ω, (S e , e ∈ E), (C v , v ∈ V )) is defined to be a graphical realization of a code C, or simply a realization of C on G, if B| I = C. A graphical realization of C in which the underlying graph G is a tree, is called a tree realization of C. Our definition of a graphical (and tree) realization differs slightly from the prior definitions in [6], [7], [8], [11], in that we require the underlying graphical model to be essential. As explained above, any graph model can be essentialized, so there is no loss of generality in this definition.…”

“…This minimal tree realization, which we henceforth denote by M(C; T, ω), is unique up to isomorphism 5 . Constructions of M(C; T, ω) can be found in [6], [7], [11]. It has further been shown [11] that not only does M(C; T, ω) minimize (among realizations in R(C; T, ω)) the state space dimension at each edge of T , but it also minimizes the local constraint code dimension at each vertex of T .…”

“…It is known that among all tree realizations of a code C that extend a given tree decomposition, there is one that minimizes the dimension of the state space at each edge of the underlying tree, and this minimal tree realization is unique [6]. It has further been shown [11] that this unique minimal tree realization also minimizes (among all tree realizations extending the given tree decompositions) the dimension of the local constraint code at each vertex of the tree. In particular, it has the least κ-complexity among all such tree realizations.…”