Local Irreducibility of Tail-Biting Trellises

This paper investigates tail-biting trellis realizations for linear block codes. Intrinsic trellis properties are used to characterize irreducibility on given intervals of the time axis. It proves beneficial to always consider the trellis and its dual simultaneously. A major role is played by trellis properties that amount to observability and controllability for fragments of the trellis of various lengths. For fragments of length less than the minimum span length of the code it is shown that fragment observability and fragment controllability are equivalent to irreducibility. For reducible trellises, a constructive reduction procedure is presented. The considerations also lead to a characterization for when the dual of a trellis allows a product factorization into elementary ("atomic") trellises

Unique Factorization and Controllability of Tail-Biting Trellis Realizations via Controller Granule Decompositions

The Conti-Boston factorization theorem (CBFT) for linear tail-biting trellis realizations is extended to group realizations with a new and simpler proof, based on a controller granule decomposition of the behavior and known controllability results for group realizations. Further controllability results are given; e.g., a trellis realization is controllable if and only if its top (controllability) granule is trivial.Comment: 5 pages, 2 figures. To be presented at the IEEE Information Theory Workshop, Jerusalem, April 201

Observability, Controllability and Local Reducibility of Linear Codes on Graphs

This paper is concerned with the local reducibility properties of linear realizations of codes on finite graphs. Trimness and properness are dual properties of constraint codes. A linear realization is locally reducible if any constraint code is not both trim and proper. On a finite cycle-free graph, a linear realization is minimal if and only if every constraint code is both trim and proper. A linear realization is called observable if it is one-to-one, and controllable if all constraints are independent. Observability and controllability are dual properties. An unobservable or uncontrollable realization is locally reducible. A parity-check realization is uncontrollable if and only if it has redundant parity checks. A tail-biting trellis realization is uncontrollable if and only if its trajectories partition into disconnected subrealizations. General graphical realizations do not share this property.Comment: 5 pages; submitted to the 2012 IEEE International Symposium on Information Theor

Characteristic Matrices and Trellis Reduction for Tail-Biting Convolutional Codes

Basic properties of a characteristic matrix for a tail-biting convolutional code are investigated. A tail-biting convolutional code can be regarded as a linear block code. Since the corresponding scalar generator matrix Gt has a kind of cyclic structure, an associated characteristic matrix also has a cyclic structure, from which basic properties of a characteristic matrix are obtained. Next, using the derived results, we discuss the possibility of trellis reduction for a given tail-biting convolutional code. There are cases where we can find a scalar generator matrix Gs equivalent to Gt based on a characteristic matrix. In this case, if the polynomial generator matrix corresponding to Gs has been reduced, or can be reduced by using appropriate transformations, then trellis reduction for the original tail-biting convolutional code is realized. In many cases, the polynomial generator matrix corresponding to Gs has a monomial factor in some column and is reduced by dividing the column by the factor. Note that this transformation corresponds to cyclically shifting the associated code subsequence (a tail-biting path is regarded as a code sequence) to the left. Thus if we allow partial cyclic shifts of a tail-biting path, then trellis reduction is accomplished.Comment: 25 pages, 3 figure

Algebraic Construction of Tail-Biting Trellises for Linear Block Codes

In this paper, we present an algebraic construction of tail-biting trellises. The proposed method is based on the state space expressions, i.e., the state space is the image of the set of information sequences under the associated state matrix. Then combining with the homomorphism theorem, an algebraic trellis construction is obtained. We show that a tail-biting trellis constructed using the proposed method is isomorphic to the associated Koetter-Vardy (KV) trellis and tail-biting Bahl-Cocke-Jelinek-Raviv (BCJR) trellis. We also evaluate the complexity of the obtained tail-biting trellises. On the other hand, a matrix consisting of linearly independent rows of the characteristic matrix is regarded as a generalization of minimal-span generator matrices. Then we show that a KV trellis is constructed based on an extended minimal-span generator matrix. It is shown that this construction is a natural extension of the method proposed by McEliece (1996).Comment: 14 pages, 10 figure