Minimal Tail-Biting Trellises for Certain Cyclic Block Codes Are Easy to Construct

Shankar, Priti; Kumar, P. N. A.; Singh, Harmeet; Rajan, B. Sundar

doi:10.1007/3-540-48224-5_52

Cited by 4 publications

(6 citation statements)

References 13 publications

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“…It is known that any optimal SISO decoder for the [31, 21, 5] BCH code must contain hidden variables with alphabet size at least 1024 [13]. Furthermore, it is known that under the standard cyclic bit ordering, the minimal tail-biting trellis for this code also must contain 1024-ary hidden variables [14]. It is thus remarkable that the RRD and cyclic-RRD decoders, which contain only binary variables, perform only 0.25 and 0.5 dB worse than the optimal SISO decoder at a BER of 10 −5 .…”

Section: B the [31 21 5] And [63 39 9] Bch Codesmentioning

confidence: 99%

Random Redundant Soft-In Soft-Out Decoding of Linear Block Codes

Halford

Chugg

2006

2006 IEEE International Symposium on Information Theory

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A number of authors have recently considered iterative soft-in soft-out (SISO) decoding algorithms for classical linear block codes that utilize redundant Tanner graphs. Jiang and Narayanan presented a practically realizable algorithm that applies only to cyclic codes while Kothiyal et al. presented an algorithm that, while applicable to arbitrary linear block codes, does not imply a low-complexity implementation. This work first presents the aforementioned algorithms in a common framework and then presents a related algorithm-random redundant iterative decoding-that is both practically realizable and applicable to arbitrary linear block codes. Simulation results illustrate the successful application of the random redundant iterative decoding algorithm to the extended binary Golay code. Additionally, the proposed algorithm is shown to outperform Jiang and Narayanan's algorithm for a number of Bose-Chaudhuri-Hocquenghem (BCH) codes.

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Section: B the [31 21 5] And [63 39 9] Bch Codesmentioning

confidence: 99%

Random Redundant Soft-In Soft-Out Decoding of Linear Block Codes

Halford

Chugg

2006

2006 IEEE International Symposium on Information Theory

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“…For nonconventional trellises the situation is not as easy, and we will go back to that in Section 5. For much on minimal conventional trellises see the comprehensive survey [36], while for the nonconventional case see [2,5,24,31,32,34].…”

Section: Minimal Trellisesmentioning

confidence: 99%

“…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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“…Then from the BCJR construction, we know that a conventional trellis for is completely specified by the sequence where (10) where we recall that the column-space of each represents the set of states at level of the trellis. Every nonmergeable linear tail-biting trellis for can be specified as where (11) Define the rank of the sequence as follows: rank rank (12) Thus the -minimal tail-biting trellis computation problem may thus be formulated as follows.…”

Section: Lemma 11: the Class Of Trellises Computed By The T-bcjr Consmentioning

confidence: 99%

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

Nori

Shankar

2006

IEEE Trans. Inform. Theory

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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

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Minimal Tail-Biting Trellises for Certain Cyclic Block Codes Are Easy to Construct

Abstract: Abstract. We give simple algorithms for the construction of generator matrices for minimal tail-biting trellises for a powerful and practical subclass of the linear cyclic codes, from which the combinatorial representation in the form of a graph can be obtained by standard procedures.

Cited by 4 publications

References 13 publications

Random Redundant Soft-In Soft-Out Decoding of Linear Block Codes

Random Redundant Soft-In Soft-Out Decoding of Linear Block Codes

On the Algebraic Structure of Linear Tail-Biting Trellises

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

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