7 research outputs found
Linear tail-biting trellises: Characteristic generators and the BCJR-construction
We investigate the constructions of tail-biting trellises for linear block
codes introduced by Koetter/Vardy (2003) and Nori/Shankar (2006). For a given
code we will define the sets of characteristic generators more generally than
by Koetter/Vardy and we will investigate how the choice of characteristic
generators affects the set of resulting product trellises, called KV-trellises.
Furthermore, we will show that each KV-trellis is a BCJR-trellis, defined in a
slightly stronger sense than by Nori/Shankar, and that the latter are always
non-mergeable. Finally, we will address a duality conjecture of Koetter/Vardy
by making use of a dualization technique of BCJR-trellises and prove the
conjecture for minimal trellises.Comment: 28 page
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
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
Codes on Graphs: Observability, Controllability and Local Reducibility
This paper investigates properties of realizations of linear or group codes
on general graphs that lead to local reducibility.
Trimness and properness are dual properties of constraint codes. A linear or
group realization with a constraint code that is not both trim and proper is
locally reducible. A linear or group realization on a finite cycle-free graph
is minimal if and only if every local constraint code is trim and proper.
A realization is called observable if there is a one-to-one correspondence
between codewords and configurations, and controllable if it has independent
constraints. A linear or group realization is observable if and only if its
dual is controllable. A simple counting test for controllability is given. An
unobservable or uncontrollable realization is locally reducible. Parity-check
realizations are controllable if and only if they have independent parity
checks. In an uncontrollable tail-biting trellis realization, the behavior
partitions into disconnected subbehaviors, but this property does not hold for
non-trellis realizations. On a general graph, the support of an unobservable
configuration is a generalized cycle.Comment: 16 pages. To appear in the IEEE Transactions on Information Theor
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
Codes on Graphs: Observability, Controllability, and Local Reducibility
Original manuscript: August 30, 2012This paper investigates properties of realizations of linear or group codes on general graphs that lead to local reducibility. Trimness and properness are dual properties of constraint codes. A linear or group realization with a constraint code that is not both trim and proper is locally reducible. A linear or group realization on a finite cycle-free graph is minimal if and only if every local constraint code is trim and proper. A realization is called observable if there is a one-to-one correspondence between codewords and configurations, and controllable if it has independent constraints. A linear or group realization is observable if and only if its dual is controllable. A simple counting test for controllability is given. An unobservable or uncontrollable realization is locally reducible. Parity-check realizations are controllable if and only if they have independent parity checks. In an uncontrollable tail-biting trellis realization, the behavior partitions into disconnected sub-behaviors, but this property does not hold for nontrellis realizations. On a general graph, the support of an unobservable configuration is a generalized cycle
Codes on Graphs: Duality and MacWilliams Identities
A conceptual framework involving partition functions of normal factor graphs is introduced, paralleling a similar recent development by Al-Bashabsheh and Mao. The partition functions of dual normal factor graphs are shown to be a Fourier transform pair, whether or not the graphs have cycles. The original normal graph duality theorem follows as a corollary. Within this framework, MacWilliams identities are found for various local and global weight generating functions of general group or linear codes on graphs; this generalizes and provides a concise proof of the MacWilliams identity for linear time-invariant convolutional codes that was recently found by Gluesing-Luerssen and Schneider. Further MacWilliams identities are developed for terminated convolutional codes, particularly for tail-biting codes, similar to those studied recently by Bocharova, Hug, Johannesson, and Kudryashov