Lower Bounds on the State Complexity of Linear Tail-Biting Trellises

Shany, Yaron; Reuven, I.; Be'ery, Y.

doi:10.1109/tit.2004.825402

Cited by 1 publication

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“…Steady subsequent research (e.g. [3,14,18,19,30,31,32,33,34]) on the top of the seminal works has led to a fairly rich development of linear trellis theory.…”

Section: Introductionmentioning

confidence: 99%

“…This groundbreaking result enabled them to achieve in [24] breakthrough on the minimality problem (which is far more complicated in the general case than in the conventional case) by narrowing down the search for minimal linear trellises to computable characteristic sets of elementary trellises, which inspired much of the subsequent research.Steady subsequent research (e.g. [3,14,18,19,30,31,32,33,34]) on the top of the seminal works has led to a fairly rich development of linear trellis theory.However, some important foundational questions have not been addressed, and as a consequence the problem of classifying minimal linear trellis representations has been addressed only partially. This problem is important not only for theoretical purposes but also for iterative/LP decoding (as we point out in Subsection 5.5).In this paper we build an algebraic framework that gives extra insight into linear trellises, answers such fundamental questions, and provides new algebraic tools that we apply to address the classification problem and more.…”

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confidence: 99%

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On the Algebraic Structure of Linear Tail-Biting Trellises

Conti¹,

Boston

2015

IEEE Trans. Inform. Theory

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Trellises are crucial graphical representations of codes. While conventional trellises are well understood, the general theory of (tail-biting) trellises is still under development. Iterative decoding concretely motivates such theory. In this paper we first develop a new algebraic framework for a systematic analysis of linear trellises which enables us to address open foundational questions. In particular, we present a useful and powerful characterization of linear trellis isomorphy. We also obtain a new proof of the Factorization Theorem of Koetter/Vardy and point out unnoticed problems for the group case.Next, we apply our work to: describe all the elementary trellis factorizations of linear trellises and consequently to determine all the minimal linear trellises for a given code; prove that nonmergeable one-to-one linear trellises are strikingly determined by the edge-label sequences of certain closed paths; prove self-duality theorems for minimal linear trellises; analyze quasi-cyclic linear trellises and consequently extend results on reduced linear trellises to nonreduced ones. To achieve this, we also provide new insight into mergeability and path connectivity properties of linear trellises.Our classification results are important for iterative decoding as we show that minimal linear trellises can yield different pseudocodewords even if they have the same graph structure. decoding (sparked by Wiberg's thesis [37] and the invention of Turbo Codes) as its complexity benefits from the long known fact proven in [35] that nonconventional representations can achieve smaller size (while optimal decoding does not, see for example [31]).The first works [5,22,23,24] towards such general theory provided a rigorous basis to the subject and had a strong influence on what came next. In particular, Koetter/Vardy [23] considered the trellis product operation (introduced first for conventional trellises in [26] and then extended to all trellises in [5]) and proved that all linear trellises factor as products of elementary trellises (Factorization Theorem), which can be more easily handled. This groundbreaking result enabled them to achieve in [24] breakthrough on the minimality problem (which is far more complicated in the general case than in the conventional case) by narrowing down the search for minimal linear trellises to computable characteristic sets of elementary trellises, which inspired much of the subsequent research.Steady subsequent research (e.g. [3,14,18,19,30,31,32,33,34]) on the top of the seminal works has led to a fairly rich development of linear trellis theory.However, some important foundational questions have not been addressed, and as a consequence the problem of classifying minimal linear trellis representations has been addressed only partially. This problem is important not only for theoretical purposes but also for iterative/LP decoding (as we point out in Subsection 5.5).In this paper we build an algebraic framework that gives extra insight into linear trellises, answers such fundamental questio...

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“…Steady subsequent research (e.g. [3,14,18,19,30,31,32,33,34]) on the top of the seminal works has led to a fairly rich development of linear trellis theory.…”

Section: Introductionmentioning

confidence: 99%