Unifying Views of Tail-Biting Trellis Constructions for Linear Block Codes

Abstract

In this paper, we present new ways of describing and constructing linear tail-biting trellises for block codes. We extend the well-known Bahl-Cocke-Jelinek-Raviv (BCJR) construction for conventional trellises to tail-biting trellises. The BCJR-like labeling scheme yields a simple specification for the tail-biting trellis for the dual code, with the dual trellis having the same state-complexity profile as that of the primal code. Finally, we show that the algebraic specification of Forney for state spaces of conventional trellises has a natural extension to tail-biting trellises. Index Terms-Block codes, displacement matrix, duality, linear trellises, nonmergeable trellises, tail-biting trellises. I. INTRODUCTION T AIL-BITING trellises for linear block codes represent the simplest form of factor graphs with cycles. These regularly structured graphs are viewed as trellises on a circular time axis and are known to attain a state-complexity that could be considerably lower than that of a conventional trellis for the same code. Conventional trellises for block codes have a well-understood underlying theory [1]-[7] and an excellent survey appears in [7]. The theory of tail-biting trellises appears to be somewhat more involved, though several advances in the understanding of the structure and properties of such trellises have been made in recent years [8]-[14]. Given a block code, it is known that there exists a unique minimal conventional trellis representing the code [6], and there are several different algorithms for the construction of such a trellis [1]-[3], [5]. The trellis simultaneously minimizes all measures of minimality. However, it is known that tail-biting trellises do not have such a property. Koetter and Vardy [9], [10] have made a detailed study of the structure of linear tail-biting trellises and have also defined several measures of minimality. They have shown that different measures of minimality correspond to different partial orders on the set of linear tail-biting trellises for a block code. An interesting property that is known for conventional trellises is that the minimal conventional trellis for a linear block Manuscript

Tail-biting products trellises, the BCJR-construction and their duals

“…These trellises were introduced in [12] and generalize the conventional BCJR-construction of minimal trellises as introduced by [1], see also [9]. We investigate the relation to product trellises, and in particular, we show that every KV-trellis is a BCJR-trellis (in our sense, which is slightly stronger than in [12]). Moreover, BCJRtrellises are nonmergeable and hence so are KV-trellises.…”

“…We will discuss the relation between these two approaches and present further properties. We will show that BCJR-trellises (in a slightly stronger sense than in [12]) are nonmergeable and that all (one-to-one) product trellises that arise from the characteristic matrix defined in [6] are BCJR-trellises and hence are nonmergeable. Finally, we will discuss a duality conjecture stated in [6].…”

“…In [6], Koetter/Vardy gave a construction from which all minimal tail-biting trellises can be derived, but this construction also generates certain non-minimal trellises. In [12], Nori/Shankar presented a construction of tail-biting trellises which generalizes the well-known BCJR-construction of conventional trellises. We will discuss the relation between these two approaches and present further properties.…”

“…. , n − 1}, on which we will compute modulo n. For details see, e. g. [2], [6], [12]. The trellis will be called conventional if the vertex space V 0 is a singleton.…”

“…We will show that each one-to-one product trellis can be merged to a BCJR-trellis defined in a slightly stronger sense than in [12] and that each trellis that originates from the characteristic matrix defined in [6] is a BCJR-trellis. Furthermore, BCJR-trellises are always nonmergeable.…”