Matrix Theory for Minimal Trellises

TL;DR

This paper introduces a unique reduced minimal span form and characteristic matrix for matrices, revealing new properties and dualities that clarify the structure of minimal trellises and their constructions.

Contribution

It defines a unique reduced minimal span form and characteristic matrix, establishing duality conditions and properties that unify different trellis construction methods.

Findings

Characteristic matrices are in duality iff their column spaces are orthogonal.

The transpose of a characteristic matrix is characteristic iff it is reduced.

Confirmed Koetter and Vardy's conjecture on lexicographical ordering in dual matrices.

Abstract

Trellises provide a graphical representation for the row space of a matrix. The product construction of Kschischang and Sorokine builds minimal conventional trellises from matrices in minimal span form. Koetter and Vardy showed that minimal tail-biting trellises can be obtained by applying the product construction to submatrices of a characteristic matrix. We introduce the unique reduced minimal span form of a matrix and we obtain an expression for the unique reduced characteristic matrix. Among new properties of characteristic matrices we prove that characteristic matrices are in duality if and only if they have orthogonal column spaces, and that the transpose of a characteristic matrix is again a characteristic matrix if and only if the characteristic matrix is reduced. These properties have clear interpretations for the unwrapped unit memory convolutional code of a tail-biting trellis, they explain the duality for the class of Koetter and Vardy trellises, and they give a natural relation between the characteristic matrix based Koetter-Vardy construction and the displacement matrix based Nori-Shankar construction. For a pair of reduced characteristic matrices in duality, one is lexicographically first in a forward direction and the other is lexicographically first in the reverse direction. This confirms a conjecture by Koetter and Vardy after taking into account the different directions for the lexicographical ordering.