A.R. Calderbank, G.D. Forney, Jr., and A. Vardy, "Minimal tail-biting trellises: the Golay code and more," IEEE Trans. Inform. Theory, to appear, July 1999.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....when dealing with the parity space given by (10) As a result, we get only about a 15 improvement in complexity compared with trellis decoding of [20] At this stage, we consider another interesting generator matrix of the (48; 24; 12) code. The theory of tail biting trellises has been given in [3]. A tail biting 16 state trellis representation for the (24; 12; 8) Golay code G 24 was given in [3] and this can be used to construct a Type II convolutional code. Following this work, Koetter and Vardy constructed a tail biting generator matrix GKV for the (48; 24; 12) code which appeared in ....

....in complexity compared with trellis decoding of [20] At this stage, we consider another interesting generator matrix of the (48; 24; 12) code. The theory of tail biting trellises has been given in [3] A tail biting 16 state trellis representation for the (24; 12; 8) Golay code G 24 was given in [3], and this can be used to construct a Type II convolutional code. Following this work, Koetter and Vardy constructed a tail biting generator matrix GKV for the (48; 24; 12) code which appeared in [18] This generator matrix has been used to construct a Type II convolutional code [18] Although GKV ....

A. R. Calderbank, G. D. Forney, Jr., and A. Vardy, "Minimal tail-biting trellises: The Golay code and more," IEEE Trans. Infor. Theory, Vol. 45, July 1999, pp. 1435--1455.

.... the Gilbert Varshamov bound [14] With restrictions on the parameters, quasicyclic codes have been investigated in [1, 7, 8, 9, 10, 11, 19, 21, 22, 24, 30] Quasi cyclic codes have been studied in terms of circulant matrices in [12] and [13] There has been a renewed interest in quasi cyclic codes [3, 5, 6, 15, 23] due to their close relationship with tail biting representations of general block # This work was partly supported by CSIR, India, through Research Grant (22(0298) 99 EMR II) to B. S. Rajan Part of this work was presented in ICCCD 2000, Kharagpur, India and ISIT 2001, Washington D.C. USA codes ....

....due to their close relationship with tail biting representations of general block # This work was partly supported by CSIR, India, through Research Grant (22(0298) 99 EMR II) to B. S. Rajan Part of this work was presented in ICCCD 2000, Kharagpur, India and ISIT 2001, Washington D.C. USA codes [3]. For instance, motivated by the 64 state quasi cyclic representation of the (24, 12, 8) Golay code, reported in [20] the theory of tail biting representation of block codes was initiated in [3] and the minimal tail biting trellises for several codes including the Golay code were reported. For ....

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Calderbank, A.R., Forney, G.D., Vardy, A.: Minimal Tail-biting Trellises: The Golay Codes and More. IEEE trans. Inform. Theory 45, 1435--1255 (1999)

....accepting path in such an automaton (where transitions are assigned costs based on a channel model. However, trellises for many useful block codes are often too large to be of practical value. Of immense interest therefore, are tail biting trellises for block codes, recently introduced in [2], which have reduced state complexity. The strings accepted by a finite state machine represented by a trellis are all of the same length, that is the block length of the code. Coding theorists therefore attach to all states that can be reached by strings of the same length l,atime index l. ....

....well known that for conventional trellises representing linear block codes, there is a unique minimal trellis for a given linear block code. However this is not true in general for tail biting trellises. There are three notions of minimality that have been suggested for tail biting trellises. In [2] a trellis is called minimal if Smax is minimized over all possible permutations of the time index, and choices of generator matrices. A second definition is # minimality, also defined in [2] where the goal is to minimize the product of all state space sizes over all permutations of the time ....

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A.R.Calderbank, G.David Forney,Jr., and Alexander Vardy, Minimal Tail-Biting Trellises: The Golay Code and More, IEEE Trans. Inform. Theory 45(5) July 1999, pp 1435-1455.

....cheapest accepting path in such an automaton(where transitions are assigned costs based on a channel model) However, trellises for many useful block codes are often too large to be of practical value. Of immense interest therefore, are tail biting trellises for block codes, recently introduced in [3], S. Yu and A. Paun (Eds. CIAA 2000, LNCS 2088, pp. 279 291, 2001. Springer Verlag Berlin Heidelberg 2001 280 P. Shankar et al. which have reduced state complexity. The strings accepted by a finite state machine represented by a trellis are all of the same length, that is the block length ....

....simulates the decoder on the overlayed automaton and outputs the decoded vector and other statistics for the range of signal to noise ratios(SNR) requested by the user. Simulations on the hexacode[5] and on the practically important Golay code, the tailbiting trellises of which are both available[3], indicate that there is a significant gain in decoding rate using the new algorithm on the tailbiting trellis over the Viterbi algorithm on the conventional trellis. We hope to augment the package with a module to convert from a minimal conventional trellis to a minimal tailbiting trellis when ....

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A.R.Calderbank, G.David Forney,Jr., and Alexander Vardy, Minimal Tail-Biting Trellises: The Golay Code and More, IEEE Trans. Inform. Theory 45(5) July 1999,pp 1435-1455.

....superposition of smaller trellises so that some states are shared. Thirdly, not all decompositions of the original trellis allow for superposition to obtain a smaller trellis. The new trellises obtained by this procedure belong to a class termed tail biting trellises described in a recent paper [2]. This class has assumed importance in view of the fact that trellises constructed in this manner can have low state complexity when compared with equivalent conventional trellises. It has been shown [17] that the maximum of the number of states in a tail biting trellis at any time index could be ....

.... overlay at each time index While, to the best of our knowledge, there are no published algorithms to solve these problems e#ciently, in the general case, there are several examples of constructions of minimal tailbiting trellises for specific examples from generator matrices in specific forms in [2]. In the next few paragraphs, we define an object called an overlayed trellis and examine the conditions under which it can be constructed so that it achieves certain bounds. Let C be a linear code over a finite alphabet. Actually a group code would su#ce, but all our examples are drawn from ....

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A.R.Calderbank, G.David Forney,Jr., and Alexander Vardy, Minimal Tail-Biting Trellises: The Golay Code and More, IEEE Trans. Inform. Theory 45(5) July 1999,pp 1435-1455.

....and its cosets, and derive necessary and sucient conditions for overlaying. The conditions turn out to be simple constraints on the coset leaders. The trellises obtained in this manner turn out to belong to the class of tailbiting trellises for block codes, recently described in the literature[3]. Section 2 gives a brief review of block codes and trellises; section 3 views block codes as nite state languages and derives the conditions for overlaying; section 4 describes the decoding algorithm and presents the results of simulations on a code called the hexacode. Finally section 5 ....

....the superposition of smaller trellises so that some states are shared. Thirdly, not all decompositions of the original trellis allow superposition to obtain a smaller trellis. The new trellises 8 obtained by this procedure belong to a class termed tail biting trellises described in a recent paper [3]. This class has assumed importance in view of the fact that trellises constructed in this manner can have low state complexity when compared with equivalent conventional trellises. It has been shown [12] that the maximum of the number of states in a tail biting trellis at any time index could be ....

[Article contains additional citation context not shown here]

A.R.Calderbank, G.David Forney,Jr., and Alexander Vardy, Minimal Tail-Biting Trellises: The Golay Code and More, IEEE Trans. Inform. Theory 45(5) July 1999,pp 1435-1455.

....the analog design of our MAP decoder. Section 5 gives our simulation and testing results. Our conclusions are given in Section 6. 2: The Hamming Code and MAP Decoding Our designs derive from the minimal tail biting trellis representation of the (8, 4) Extended Hamming code, as discussed in [4]. The Hamming encoder takes a block of four information bits u and produces a block of eight digital bits x. Thus, while there are 256 possible 8 bit sequences, only 16 of them can be produced as the encoder s output. A valid output sequence is referred to as a codeword. Any two codewords are ....

....not digital. The encoder instead selects its output from a set of physical signals. A transmittable signal is represented by a distinct symbol, or label. The set of these labels is referred to as the trellis alphabet. The minimal tail biting trellis for the (8, 4) Hamming code was developed in [4] and is shown in Figure 1. It is called a tail biting graph because the nal state space is connected to the initial state space. This means that a path through the trellis is possible, or valid, only if it begins and ends in the same state. The branches in the diagram indicate allowable ....

A. R. Calderbank, G. D. Forney, Jr., and A. Vardy, Minimal Tail-Biting Trellises: The Golay Code and More, IEEE Trans. Inform. Theory, vol. 45, pp. 1435-1455, July 1999.

.... 10 0 10 5 Eb No union bound w o pseudocodewords union bound w pseudocodewords ML decoding Iterative decoding(5 iterations) Figure 1: 8,4,4) tail biting Hamming code union bound The pseudoweight enumerator for the (8; 4; 4) Hamming code, as represented by the minimal tail biting trellis from [5] is given in the table below. In the first column is the ordinary weight enumerator, i.e. a list of the weights of the codewords (the one segment pseudocodewords) In the second column is the pseudoweight enumerator for the 64 two segment pseudocodewords. There can be no simple pseudocodewords ....

A. R. Calderbank, G. D. Forney, Jr., and A. Vardy, "Minimal tail-biting trellises: the Golay code and more." submitted to IEEE Trans. Inform. Theory.

...., then x i is the least weight of any path in G 0 from vertex i to vertex 1, and y j is the least weight of any path from vertex 1 to vertex j. 3 Tail Biting Codes and Pseudo Codewords In this section we give a brief introduction to tail biting codes. For further details, we refer the reader to [8]. A tail biting trellis is a finite, labeled, digraph in which the vertices are partitioned into n classes, Sigma 0 ; Sigma n Gamma1 , each class being indexed by an element of Z n = f0; 1; n Gamma 1g, the cyclic group of order n. All index arithmetic is done modulo n. If E is ....

.... union bound w o pseudocodewords union bound w pseudocodewords ML decoding Iterative decoding(5 iterations) Figure 1: 8,4,4) tail biting Hamming code union bound on AWGN channel The pseudoweight enumerator for the (8; 4; 4) Hamming code, as represented by the minimal tail biting trellis from [8] is given in the table below. In the first row is the ordinary weight enumerator, i.e. a list of the weights of the codewords (the one segment pseudocodewords) In the second row is the pseudoweight enumerator for the 64 twosegment pseudocodewords. There can be no simple pseudocodewords with ....

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A. R. Calderbank, G. D. Forney, Jr., and A. Vardy, "Minimal tail-biting trellises: the Golay code and more." submitted to IEEE Trans. Inform. Theory.

....study the approximate correctness of PPA on graphs with cycles. In this paper we make a first step by discussing the behavior of an PPA in graphs with a single cycle. This work is directly relevant to the study of iterative decoding of tail biting codes, whose underlying graph has just one cycle [3], 12] First, we shall show that for strictly positive local kernels, the iterations of the PPA will always converge to the same fixed point regardless of the scheduling order used. Moreover, the length of the cycle does not play a role in this convergence. Secondly, we shall generalize a result ....

A. R. Calderbank, G. D. Forney, Jr. and A. Vardy, "Minimal tail-biting trellises: the Golay code and more," submitted to IEEE Trans. Inform. Theory, August 1997.

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A.R. Calderbank, G.D. Forney, Jr., and A. Vardy, "Minimal tail-biting trellises: the Golay code and more," IEEE Trans. Inform. Theory, to appear, July 1999.

....of pseudocodewords for binary codewords on an AWGN channel. This paper extends Wiberg s formula for AWGN channels to nonbinary codes, develops similar results for BSC and BEC channels, and gives upper and lower bounds on the e ective weight. The 16 state tail biting trellis of the Golay code [2] is used for examples. Although in this case no pseudocodeword is found with e ective weight less than the minimum Hamming weight of the Golay code on an AWGN channel, it is shown by example that the minimum e ective pseudocodeword weight can be less than the minimum codeword weight. Key words. ....

....nonbinary case; 2. We give lower and upper bounds on w e ; 3. We develop similar results for the binary symmetric channel (BSC) and binary erasure channel (BEC) As examples, we compute the e ective weights of certain pseudocodewords in the 16 state TBT of the binary (24, 12, 8) Golay code of [2]. For the AWGN channel, we have not found any examples of pseudocodewords with e ective weight less than 8; however, neither have we been able to prove that 8 is the minimum e ective weight for this case. For the binary symmetric channel, on the other hand, we exhibit a pseuodcodeword with ....

A. R. Calderbank, G. D. Forney, Jr. and A. Vardy, \Minimal tail-biting trellises: The Golay code and more," IEEE Trans. Inform. Theory, vol. 45, pp. 1435{ 1455, July 1999.

....linear trellises are established. Keywords: block codes, codes on graphs, convolutional codes, linearity, linear trellises, minimal trellises, tail biting trellises, trellis complexity. 1. Introduction Trellis representations of linear block codes have received much attention in the last decade [2, 9, 5, 10, 13, 15]. Such representations illuminate code structure and often lead to efficient trellis based decoding algorithms. Today, we have not only the conventional trellises, whose theory is by now well developed [10, 13] but also the tail biting trellises. Numerous examples are known [2, 7, 11] where the ....

....[2, 9, 5, 10, 13, 15] Such representations illuminate code structure and often lead to efficient trellis based decoding algorithms. Today, we have not only the conventional trellises, whose theory is by now well developed [10, 13] but also the tail biting trellises. Numerous examples are known [2, 7, 11] where the complexity of a tailbiting trellis is much lower than the complexity of the best possible conventional trellis for the same code. Tail biting trellises for block codes were studied by several authors [2, 3, 11, 12] including McEliece [1] However, such trellises are not yet well ....

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A.R. Calderbank, G.D. Forney, Jr., and A. Vardy, "Minimal tailbiting trellises: the Golay code and more," IEEE Trans. Inform. Theory, 45, pp. 1435--1455, July 1999.

....codes, linearity, minimal trellises, tail biting trellises, trellis complexity Research supported by the David and Lucile Packard Foundation and by the National Science Foundation. 1. Introduction Trellis representations of linear block codes have received much attention in recent years [3, 13, 15, 16, 20, 31, 35]. Such representations not only illuminate code structure, but also often lead to ecient trellis based decoding algorithms. It is now well known that, given a speci c coordinate ordering, there exists a unique, up to isomorphism, minimal (conventional) trellis for any linear block code. The ....

....matrix for C . On the other hand, much less is known about tail biting trellises. Tail biting trellis representations are interesting for several reasons. First, the complexity of a tail biting trellis may be much lower than the complexity of the best possible conventional trellis. It is shown in [3, 35] that the number of states in a tail biting trellis for a linear code C can be as low as the square root of the number of states in the minimal conventional trellis for C . Secondly, tail biting trellises may be considered as the simplest form of a factor graph with cycles. The recent development ....

[Article contains additional citation context not shown here]

A.R. Calderbank, G.D. Forney, Jr., and A. Vardy, \Minimal tail-biting trellises: the Golay code and more,"IEEE Trans. Inform. Theory, vol. 45, pp. 1435-1455, July 1999.

.... , with central state space S # and with local codes Codes on Graphs: Generalized State Realizations 11 C 1 = a J A , s # ) s # # S # , a J A # C ## (s # ) and C 2 = a KA , s # ) s # # S # , a KA # C ## (s # ) From this observation follows the important Cut Set Bound [33, 5]: Theorem 4.1 (Cut Set Bound) Let I A be the symbol index set of a code C, let # be a cut set in the graph of a generalized state realization of C, and let JA and KA be the two disjoint subsets of symbol indices in the graph partition induced by removal of the edges in #. Then S # = Q ....

....given by the Cut Set Bound. It has been shown, for example, that there exists a tail biting realization of the (24, 12, 8) Golay code in which the size of each state space is only 16, whereas in any comparable conventional state realization at least one state space must have as many as 256 states [5]. 4.4 The Sum Product Algorithm The cut set idea leads to the sum product algorithm [19] an e#cient iterative algorithm for optimal decoding of cycle free generalized state realizations. The decoding problem solved by the sum product algorithm is as follows. Assume that there exists a finite ....

A. R. Calderbank, G. D. Forney, Jr. and A. Vardy, "Minimal tail-biting trellises: The Golay code and more," IEEE Trans. Inform. Theory, to appear.

....models for Boolean functions. In recent years, a number of new graphical models have emerged in coding theory, and evolved into a far reaching general framework for representing a code by a graph. In this context, one encounters various generalizations of a trellis, such as tail biting trellises [22] and trellis formations [49, 50] as well as Tanner graphs [71] that are in some sense diametrically opposite to trellises. All these representations are special cases of the general concept of a factor graph. We refer the reader to [1, 37, 38, 79] for a detailed treatment of factor graphs and ....

....79] for a detailed treatment of factor graphs and the associated iterative manipulation algorithms: the min sum and the sum product. The success of these graphical models in coding theory and communications has been spectacular. For example, tail biting trellis representations have been found in [22, 48] for several well known codes, whose complexity is the square root of the lowest complexity achievable with the conventional minimal trellis. On a grander scale, turbo codes [8] represented by a factor graph and decoded with an iterative sum product algorithm have been shown to approach channel ....

A.R. Calderbank, G.D. Forney, Jr., and A. Vardy, "Minimal tail-biting trellises: the Golay code and more," IEEE Trans. Inform. Theory, submitted for publication.

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A. R. Calderbank, G. D. Forney, and A. Vardy, "Minimal tail- biting trellises: The Golay code and more," IEEE Trans. Inform. Theory, vol. 45, pp. 1435--1455, July 1999.

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A. R. Calderbank, G. D. Forney, and A. Vardy, "Minimal tail-biting trellises: the Golay code and more," IEEE Trans. Inform. Theory, vol. 45, no. 5, pp. 1435-1455, July 1999.

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Calderbank, A. R., Forney, G. D., and Vardy, A. Minimal tail-biting trellises: The golay code and more. IEEE Trans. Inform. Theory (1999), 1435--1455.

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A. R. Calderbank, G. D. Forney, and A. Vardy, "Minimal tail-biting trellises: The Golay code and more," IEEE Trans. Inform. Theory, vol. 45, pp. 1435--1455, July 1999.

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