On complexity of trellis structure of linear block codes
24 Jun 1991-Vol. 39, Iss: 3, pp 1057-1064
TL;DR: An upper bound on the number of states of a minimal trellis diagram for a linear block code is derived and a cyclic code or its extended code is shown to be the worst in terms of Trellis state complexity among the linear codes of the same length and dimension.
Abstract: An upper bound on the number of states of a minimal trellis diagram for a linear block code is derived. Using this derivation a cyclic (or shortened cyclic) code or its extended code is shown to be the worst in terms of trellis state complexity among the linear codes of the same length and dimension. The complexity of the minimal trellis diagrams for linear block codes of length 2/sup m/, including the Reed-Muller codes, is analyzed. The construction of minimal trellis diagrams for some extended and permuted primitive BCH codes is presented. It is shown that these codes have considerably simpler trellis structure than the original codes in cyclic form without bit-position permutation. >
Citations
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27 Jun 1994
TL;DR: A novel approach to soft decision decoding for binary linear block codes that achieves a desired error performance progressively in a number of stages and is terminated at the stage where either near-optimum error performance or a desired level of error performance is achieved.
Abstract: Presents a novel approach to soft decision decoding for binary linear block codes. The basic idea is to achieve a desired error performance progressively in a number of stages. For each decoding stage, the error performance is tightly bounded and the decoding is terminated at the stage where either near-optimum error performance or a desired level of error performance is achieved. As a result, more flexibility in the tradeoff between performance and decoding complexity is provided. The decoding is based on the reordering of the received symbols according to their reliability measure. The statistics of the noise after ordering are evaluated. Based on these statistics, two monotonic properties which dictate the reprocessing strategy are derived. Each codeword is decoded in two steps: (1) hard-decision decoding based on reliability information and (2) reprocessing of the hard-decision-decoded codeword in successive stages until the desired performance is achieved. The reprocessing is based on the monotonic properties of the ordering and is carried out using a cost function. A new resource test tightly related to the reprocessing strategy is introduced to reduce the number of computations at each reprocessing stage. For short codes of lengths N/spl les/32 or medium codes with 32 >
636 citations
Book•
09 Sep 2002
TL;DR: This paper presents a Comparative Study of Turbo Equalisers: The Super Trellis Structure of Convolutional Turbo Codes and the Coded Modulation Theory and Performance.
Abstract: Acknowledgments.Historical Perspective, Motivation and Outline. I Convolutional and Block Coding. Convolutional Channel Coding. Block Coding. Soft Decoding and Performance of BCH Codes. II Turbo Convolutional and Turbo Block Coding. Turbo Convolutional Coding. The Super Trellis Structure of Convolutional Turbo Codes. Turbo BCH Coding. Redundant Residue Number System Codes. III Coded Modulation: TCM, TTCM, BICM, BICM ID. Coded Modulation Theory and Performance. IV Space Time Block and Space Time Trellis Coding. Space time Block Codes. Space Time Trellis Codes. Turbo coded Adaptive QAM versus Space time Trellis Coding. V Turbo Equalisation. Turbo coded Partial response Modulation. Turbo Equalisation for Partial response Systems. Turbo Equalisation Performance Bound. Comparative Study of Turbo Equalisers. Reduced complexity Turbo Equaliser. Turbo Equalisation for Space time Trellis coded Systems. Summary and Conclusions. Bibliography. Subject Index. Author Index. About the Authors.Other Related Wiley and IEEE Press Books.
407 citations
Book•
01 Aug 2002
TL;DR: This book endeavours to be the first book with explicit emphasis on channel coding for transmission over wireless channels, and systematically converts the lessons of Shannon's information theory into design principles applicable to practical wireless systems in a comprehensive manner.
Abstract: Against the backdrop of the emerging 3G wireless personal communications standards and broadband access network standard proposals, this volume covers a range of coding and transmission aspects for transmission over fading wireless channels. It presents the most important classic channel coding issues and also the exciting advances of the last decade, such as turbo coding, turbo equalisation and space-time coding. It endeavours to be the first book with explicit emphasis on channel coding for transmission over wireless channels. Divided into 4 parts: Part 1 - explains the necessary background for novices. It aims to be both an easy reading text book and a deep research monograph. Part 2 - provides detailed coverage of turbo conventional and turbo block coding considering the known decoding algorithms and their performance over Gaussian as well as narrowband and wideband fading channels. Part 3 - comprehensively discusses both space-time block and space-time trellis coding for the first time in literature. Part 4 - provides an overview of turbo equalisations, also referred to as turbo demodulation. The book systematically converts the lessons of Shannon's information theory into design principles applicable to practical wireless systems. It provides overall design performance studies, giving cognizance to the contradictory design requirements of bit error rate, implementational complexity, coding and interleaving delay, effective throughput, coding rate and other related systems design aspects in a comprehensive manner.
303 citations
TL;DR: If C is complete and strongly controllable, then a minimal encoder in controller canonical (feedbackfree) form may be constructed from certain sets of shortest possible code sequences, called granules.
Abstract: A group code C over a group G is a set of sequences of group elements that itself forms a group under a component-wise group operation. A group code has a well-defined state space Sigma /sub k/ at each time k. Each code sequence passes through a well-defined state sequence. The set of all state sequences is also a group code, the state code of C. The state code defines an essentially unique minimal realization of C. The trellis diagram of C is defined by the state code of C and by labels associated with each state transition. The set of all label sequences forms a group code, the label code of C, which is isomorphic to the state code of C. If C is complete and strongly controllable, then a minimal encoder in controller canonical (feedbackfree) form may be constructed from certain sets of shortest possible code sequences, called granules. The size of the state space Sigma /sub k/ is equal to the size of the state space of this canonical encoder, which is given by a decomposition of the input groups of C at each time k. If C is time-invariant and nu -controllable, then mod Sigma /sub k/ mod = Pi /sub 1 >
270 citations
Cites background from "On complexity of trellis structure ..."
...A partial bibliography of earlier related work would include: 1) Convolutional codes and linear systems: [ 11-[SI; 2) Trellis structure of linear codes: [9]-[ 12 ]; 3) Euclidean-space codes based on group structure:...
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TL;DR: It is shown that, among all trellises that represent a given code, the original trellis introduced by Bahl, Cocke, Jelinek, and Raviv in 1974, and later rediscovered by Wolf, Massey, and Forney, uniquely minimizes the edge count.
Abstract: In this semi-tutorial paper, we will investigate the computational complexity of an abstract version of the Viterbi algorithm on a trellis, and show that if the trellis has e edges, the complexity of the Viterbi algorithm is /spl Theta/(e). This result suggests that the "best" trellis representation for a given linear block code is the one with the fewest edges. We will then show that, among all trellises that represent a given code, the original trellis introduced by Bahl, Cocke, Jelinek, and Raviv in 1974, and later rediscovered by Wolf (1978), Massey (1978), and Forney (1988), uniquely minimizes the edge count, as well as several other figures of merit. Following Forney and Kschischang and Sorokine (1995), we will also discuss "trellis-oriented" or "minimal-span" generator matrices, which facilitate the calculation of the size of the BCJR trellis, as well as the actual construction of it.
252 citations
References
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Book•
01 Jan 1977
TL;DR: This book presents an introduction to BCH Codes and Finite Fields, and methods for Combining Codes, and discusses self-dual Codes and Invariant Theory, as well as nonlinear Codes, Hadamard Matrices, Designs and the Golay Code.
Abstract: Linear Codes. Nonlinear Codes, Hadamard Matrices, Designs and the Golay Code. An Introduction to BCH Codes and Finite Fields. Finite Fields. Dual Codes and Their Weight Distribution. Codes, Designs and Perfect Codes. Cyclic Codes. Cyclic Codes: Idempotents and Mattson-Solomon Polynomials. BCH Codes. Reed-Solomon and Justesen Codes. MDS Codes. Alternant, Goppa and Other Generalized BCH Codes. Reed-Muller Codes. First-Order Reed-Muller Codes. Second-Order Reed-Muller, Kerdock and Preparata Codes. Quadratic-Residue Codes. Bounds on the Size of a Code. Methods for Combining Codes. Self-dual Codes and Invariant Theory. The Golay Codes. Association Schemes. Appendix A. Tables of the Best Codes Known. Appendix B. Finite Geometries. Bibliography. Index.
10,083 citations
"On complexity of trellis structure ..." refers background in this paper
...5) and the property of Mattson-Solomon polynomial [5, 7]....
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...There is a permutation lr on the bit positions (see section 4) such that _r[C'] = RMm,,, the r-th order (noncyclic) Reed-Muller code of length 2" [5, 7]....
[...]
...For 0 <_ r _< m, the r-th order (noncyclic) Reed-Muller code of length 2r_ [5, 7], denoted RM"`,,, is defined as {b(f) : f E P_,}, this is, P[RM"`,,] = P_....
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...Then C' is a subcode of C and consists of the set of all binary n-tuples represented by boolean polynomials of degree 2 or less [7]....
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...Then C' is a subcode of C and consists of the set of all binary n-tuples represented by boolean polynomials of degree 3 or less [7]....
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Book•
01 Jan 1983
TL;DR: This book explains coding for Reliable Digital Transmission and Storage using Trellis-Based Soft-Decision Decoding Algorithms for Linear Block Codes and Convolutional Codes, and some of the techniques used in this work.
Abstract: 1. Coding for Reliable Digital Transmission and Storage. 2. Introduction to Algebra. 3. Linear Block Codes. 4. Important Linear Block Codes. 5. Cyclic Codes. 6. Binary BCH Codes. 7. Nonbinary BCH Codes, Reed-Solomon Codes, and Decoding Algorithms. 8. Majority-Logic Decodable Codes. 9. Trellises for Linear Block Codes. 10. Reliability-Based Soft-Decision Decoding Algorithms for Linear Block Codes. 11. Convolutional Codes. 12. Trellis-Based Decoding Algorithms for Convolutional Codes. 13. Sequential and Threshold Decoding of Convolutional Codes. 14. Trellis-Based Soft-Decision Algorithms for Linear Block Codes. 15. Concatenated Coding, Code Decomposition ad Multistage Decoding. 16. Turbo Coding. 17. Low Density Parity Check Codes. 18. Trellis Coded Modulation. 19. Block Coded Modulation. 20. Burst-Error-Correcting Codes. 21. Automatic-Repeat-Request Strategies.
3,848 citations
"On complexity of trellis structure ..." refers background in this paper
...The binary primitive BCH code of length 2" - 1 and minimum Hamming distance 2"-' - 1 contains the cyclic r-th order Reed-Muller code of length 2" - 1 as a subcode [5, 6], and the dual code of the even weight subcode of the binary primitive BCH code of length 2" - 1 and a specific designed distance, denoted q(m, r), contains the cyclic r-th order Reed-Muller code of length 2" - 1 as a subcode for q(m, r) = 5, q(m, 2) = 2 tin/2/ + 3, ....
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...Let c-RMm,, denote the cyclic r-th order Reed-Muller code of length 2"* - 1 [5, 6]....
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2,541 citations
TL;DR: This paper attempts to present a comprehensive tutorial survey of the development of efficient modulation techniques for bandlimited channels, such as telephone channels, with principal emphasis on coded modulation techniques, in which there is an explosion of current interest.
Abstract: This paper attempts to present a comprehensive tutorial survey of the development of efficient modulation techniques for bandlimited channels, such as telephone channels. After a history of advances in commercial high-speed modems and a discussion of theoretical limits, it reviews efforts to optimize two-dimensional signal constellations and presents further elaborations of uncoded modulation. Its principal emphasis, however, is on coded modulation techniques, in which there is an explosion of current interest, both for research and for practical application. Both block-coded and trellis-coded modulation are covered, in a common framework. A few new techniques are presented.
770 citations
Additional excerpts
...Our study is motivated by the works of Wolf [1] and Forney [2] [3], especially Forney's latest work [3] in which he presented a trellis construction for linear block codes and asserted that the construction results in minimal trellises in the sense of number of states....
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...Our study is motivated by the works of Wolf [1] and Forney [2, 3], especially Forney's latest work [3] in which he presented a trellis construction for linear block codes and asserted that the construction results in minimal trellises in the sense of number of states....
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...I = 2 (2.11) This was first given by Forney [3, Appendix A]....
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TL;DR: By viewing the minimum Hamming weight as a certain minimum property of one-dimensional subcodes, a generalized notion of higher-dimensional Hamming weights is obtained, which characterize the code performance on the wire-tap channel of type II.
Abstract: Motivated by cryptographical applications, the algebraic structure, of linear codes from a new perspective is studied. By viewing the minimum Hamming weight as a certain minimum property of one-dimensional subcodes, a generalized notion of higher-dimensional Hamming weights is obtained. These weights characterize the code performance on the wire-tap channel of type II. Basic properties of generalized weights are derived, the values of these weights for well-known classes of codes are determined, and lower bounds on code parameters are obtained. Several open problems are also listed. >
709 citations
"On complexity of trellis structure ..." refers background or methods in this paper
...For a binary N-tuple v, the support of v, denoted s(v), is defined as the set of indices of bit positions where the components of v are nonzero [ 7 ]....
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...antilexicographi cal order defined by XI 7 ]....
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...However, in [II], we relate this permutation problem to the generalized Hamming weight hierarchy of a code, and we show that [or Reed-Muller codes, the standard binary order of bit positions is optimum at every bit positions with respect to the state complexity of the minimal trellis diagram by using a theorem due to Wei [ 7 ]....
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