J. K. Wolf, "Efficient maximum-likelihood decoding of linear block codes using a trellis," IEEE Transactions of Information Theory, vol. 24, pp. 76-80, January 1978.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....from the parity check matrix H. Horizontal branches represent c i = 0, whereas the other branches have c i =1. The concept of trellises was originally used to decode convolutional codes [43] and the fact that block codes can be interpreted as trellises was first described in [44] and later in [45] and [46] Wolf [45] proved that a trellis for an arbitrary linear block code (N,K) over GF(q) q 2, has at most q min K,N K states and also provided a method for constructing trellises for block codes. Although Massey [46] provided a precise definition of a trellis and showed that it is ....

....matrix H. Horizontal branches represent c i = 0, whereas the other branches have c i =1. The concept of trellises was originally used to decode convolutional codes [43] and the fact that block codes can be interpreted as trellises was first described in [44] and later in [45] and [46] Wolf [45] proved that a trellis for an arbitrary linear block code (N,K) over GF(q) q 2, has at most q min K,N K states and also provided a method for constructing trellises for block codes. Although Massey [46] provided a precise definition of a trellis and showed that it is possible to design a ....

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J. K. Wolf, "Efficient maximum-likelihood decoding of linear block codes using a trellis," IEEE Transactions of Information Theory, vol. 24, pp. 76-80, January 1978.

....different code symbols. Each path from the initial state Sl to the final state St is thus a code word in C. The concept of trellises was first used to decode con volutional codes [6] and the fact that block codes can be interpreted as trellises was first described in [8] and later in [9]. The complexity of a decoding algorithm operating on the code trellis is dependent on the number of states and the number of branches. A trellis for an (n,k) block code defined over GF(q) and constructed from the gen erator matrix G will have at most qk states at any time interval, while a ....

J. K. Wolf, "Efficient maximum-likelihood decoding of linear block codes using a trellis", 1EEE Trans. Info. Tii., pp.76- 80, 1978.

....of branch symbols per information bit (See Section II D for a definition of that holds when more than 1 bit labels each branch. A. Complexity: Past Work on the Subject The literature on the trellis complexity of block codes is very rich. The subject was introduced in some classical papers [4] [8] [10] Many other papers deal with the subject, covering methods to compute the complexity parameters, build the trellis, and establish bounds, or studies on the complexity reduction by coordinate permutation. Among them we can cite [11] 26] Recent results on the complexity of convolutional ....

J. K. Wolf, "Efficient maximum likelihood decoding of linear block codes using a trellis," IEEE Trans. Inform. Theory, vol. IT-24, pp. 76--80, Jan. 1978.

.... [6, 10, 11] Block group codes constitute a basic ingredient for a large class of block coded modulation schemes [6] The coding gain achieved by these schemes is possible only with soft decision decoding [18, 17] Trellises provide a general framework for efficient soft decision decoding of codes [22], for instance using the Viterbi algorithm [3] Since the decoding effort is directly related to the size of the trellis, much work has been done on characterizing and constructing minimal trellises for group codes [16, 5, 8, 12, 13] In this paper, we present an O(k2n s) time algorithm for ....

J.K. Wolf, "Efficient maximum likelihood decoding of linear block codes using a trellis," IEEE Trans. on Inform. Theory, vol. IT-24, pp. 76-80, 1978. 153

....basis of the transformation #. Now each of these segments are of length N . We also saw that any N ary binary word can be represented as a sum of BCH codeword and a coset leader term. We can represent each of the segments using a binary trellis. The trellis is constructed by the Wolf method [10] using the parity check matrix of BCH . Such a trellis will contain paths corresponding to all possible binary N tuples. Furthermore paths corresponding to the N tuples of the same coset will terminate at a common terminal node. Fig. 1(a) shows such a trellis corresponding to the (7, 5, 3) RS ....

J. K. Wolf, "Efficient maximum likelihood decoding of linear block codes using a trellis," IEEE Trans. Inform. Theory, vol. IT-24, pp. 76--80, Jan 1978.

.... and [15] Block group codes are basic ingredients in a large class of block coded modulation schemes [7] The coding gain promised by these schemes can be achieved only with soft decision decoding, 22] and [23] Trellises provide a general framework for efficient soft decision decoding of codes [26], for instance by using the Viterbi algorithm [4] Since the decoding effort is directly related to the size of the trellis, much work has been devoted to characterizing and constructing minimal trellises for group codes [6] 9] 16] 17] and [21] In this. paper, we present an O(k2n 8) time ....

J. K. Wolf, "Efficient maximum likelihood decoding of linear block codes using a trellis," IEEE Trans. Inform. Theory, vol. IT-24, pp. 76-80, 1978.

....(DP) leads to a computationally efficient identification of the globally optimal path. Examples of applications of DP to recognition problems are numerous ad include speech recognition [1] 2] 3] character recognition [4] 5] 6] deformable template matching [7] 8] soft decoding [9] [10], 11] 12] 13] and road tracking [14] Such problems often lead to enormous state spaces, however, and the computations can be infeasible, even with DP. To overcome this obstacle, we propose a variation on DP we call coarse to fine dynamic programming (CFDP) We demonstrate two applications of ....

Wolf J. (1978), "Efficient Maximum Likelihood Decoding of Linear Block Codes Using a Trellis," IEEE Transactions on Information Theory, Vol IT-24, No. 1, 76-80.

....(DP) leads to a computationally efficient identification of the globally optimal path. Examples of applications of DP to recognition problems are numerous and include speech recognition [1] 2] 3] character recognition [4] 5] 6] deformable template matching [7] 8] soft decoding [9] [10], 11] 12] 13] and road tracking [14] Such problems often lead to enormous state spaces, however, and the computations can be infeasible, even with DP. To overcome this obstacle, we propose a variation on DP we call coarseto fine dynamic programming (CFDP) We demonstrate two applications of ....

Wolf J. (1978), "Efficient Maximum Likelihood Decoding of Linear Block Codes Using a Trellis," IEEE Transactions on Information Theory, Vol IT-24, No. 1, 76--80.

....a negligible increase in decoding error probability. Therefore we use Pmin(C) as a benchmark for all other decoding algorithms applied to code C. Similarly, decoding complexity of our design will be compared with the upper bound qmin n k,k on ML decoding complexity, which was obtained in [2] and [15] by trellis design. Given any output y, we first wish to restrict the list of inputs x X n among which the most probable codeword is being sought. More specifically, we consider the list (y, of input vectors x with maximum posterior probabilities P(x[y) among all qn inputs. In contrast to ML ....

J.K. Wolf, "Efficient maximum likelihood decoding of linear codes using a trellis," IEEE Trans. Inform. Theory, vol. 24, pp. 76-80, 1978.

....of P (v n = 0jr; v 2 V ) and the algorithm can now be described as in Figure 1. The algorithm needs one list of size 2 N GammaK and the number of operations is in the order of N2 N GammaK . The algorithm resembles soft decoding of block codes using a trellis of maximal size (unexpurgated) [8]. We compare the performance with other optimal algorithms. These optimal algorithms [6] 2] 5] have different complexity depending on the code. Algorithms described in [2] sum up the situation for binary linear block codes as follows. For K N Gamma K the 5 Input: P (r 1 jv 1 ) P (r 2 jv ....

J. K. Wolf, "Efficient maximum likelihood decoding of linear block codes using a trellis", IEEE Trans. Inform. Theory, vol IT-24, pp. 76--80, Jan. 1978.

....4) are 172 and 252 respectively and (c) the total number of computations needed for Viterbi decoding with the minimal trellises of RM(1; 4) and RM(2; 4) are 195 and 355 respectively. We now show that RM(r;m) with a total degree bit order has state complexity reaching the Wolf upper bound, [14]. 6 Corollary 2.4 If RM(r;m) has a total degree bit order, then s(RM(r;m) minfdim(r; m) dim(m Gamma r Gamma 1; m)g: Proof. Since the state complexity of a code is equal to the state complexity of its dual (e.g. 4] it is sufficient to show that for r m Gamma r Gamma 1, the state ....

Jack. K. Wolf (1978) Efficient maximum likelihood decoding of linear block codes using a trellis. IEEE Trans. Information Theory 24, 76--80. 25

.... [7, 12, 14] Block group codes constitute a basic ingredient for a large class of block coded modulation schemes [7] The coding gain achieved by these schemes is possible only with soft decision decoding [22, 21] Trellises provide a general framework for efficient soft decision decoding of codes [25], for instance using the Viterbi algorithm [4] Since the decoding effort is directly related to the size of the trellis, much work has been done on characterizing and constructing minimal trellises for group codes [20, 6, 9, 15, 16] In this paper, we present an O(k 2 n s) time algorithm for ....

J. K. Wolf, "Efficient maximum likelihood decoding of linear block codes using a trellis," IEEE Trans. on Inform. Theory, vol. IT-24, pp. 76-80, 1978.

....trellis decoding of channel codes, source information and channel characteristics. Decoding Channel Code Dependencies The inclusion of a channel code applied to the source data represents the traditional use of a Viterbi decoder. Although the Viterbi decoder can be used to decode block codes [126], only convolu tional codes are considered here. Given a convolutionally coded symbol sequence, with certain memory, we wish to decode the original symbol sequence. The memory of the code can be con sidered as introducing dependencies between the symbols of the sequence. Being a channel ....

J.K. Wolf, "Efficient Maximum Likelihood Decoding of Linear Block Codes Using a Trel- lis," IEEE Trans. Inform. Theory, vol. IT--24, pp. 76--80, Jan 1978.

....x of R n , will find a codeword u e # that minimizes dist(x , u) the distance being Euclidean distance. Soft decision decoding has been investigated by many distinguished information theorists over the years see [1] 12] 14] 31] 35] 40] 45] 51] 52] 54] 58] 60] 62] 66] [69]. However, the majority of these papers study decoding algorithms that only perform correctly most of the time. For example, Hackett s decoding algorithm [42] for the Golay code is only a few tenths of a decibel different from ideal correlation detection . In the present paper we are only ....

....[6] 7] 62] are a class of (in general) non binary codes that include first order Reed Muller codes, simplex codes, n and n as special cases, and may also be decoded rapidly. 2. 5) Codes with small redundancy may be regarded as trellis codes, as pointed out by Solomon [11] 66] and Wolf [69], and therefore can be efficiently decoded by the Viterbi algorithm [35] 68] For an [n , k] code, the trellis has 2 n k states, and the number of decoding steps is roughly 4n2 n k . 2.6) Direct sums of codes on this list can also be decoded easily. To decode , for example, we ....

J. K. Wolf, Efficient maximum likelihood decoding of linear block codes using a trellis, IEEE Trans. Information Theory, IT-24 (1978), 76-80.

....in the choice of code parameters. III Sequential Decoding of Reed Solomon Codes In this section, we describe a soft decision sequential decoder for Reed Solomon codes. Of course one approach to soft decision decoding of Reed Solomon codes is to perform a breadth4 first search of the code trellis [7] using the Viterbi algorithm. A minimal trellis (in the sense of minimizing the number of states required to represent all codewords) can be found, e.g. via Wolf s procedure [7] or in a number of other equivalent ways [8, 9, 10] Every [n; k] maximum distance separable code over GF (q) has the ....

....one approach to soft decision decoding of Reed Solomon codes is to perform a breadth4 first search of the code trellis [7] using the Viterbi algorithm. A minimal trellis (in the sense of minimizing the number of states required to represent all codewords) can be found, e.g. via Wolf s procedure [7] or in a number of other equivalent ways [8, 9, 10] Every [n; k] maximum distance separable code over GF (q) has the same minimal trellis (apart from branch labels) 10] the minimal trellis always has s i = min(q i ; q n Gammai ; q k ; q n Gammak ) states at time index i [10, 11] The ....

J. K. Wolf, "Efficient maximum-likelihood decoding of linear block codes using a trellis," IEEE Trans. on Inform. Theory, vol. IT-24, pp. 76--80, 1978.

....has a well defined and unique minimal state realization. Trellises were introduced in the coding theory literature by Forney [4] as a means of describing the Viterbi algorithm for decoding convolutional codes. Bahl, et al. 5] showed that block codes, too, can be described by a trellis, and Wolf [6] proposed the use of the Viterbi algorithm for trellis based soft decision decoding of block codes. Massey [7] gave a graph theoretic definition of a block trellis and an alternative construction for minimal trellises. Forney s celebrated paper [8] showed that group codes, including linear codes ....

....codes, lattice shells of fixed norm, and constant weight subcodes of a linear code. In Section V we show that the minimal trellis for an MFC code can be constructed as a subtrellis of the complete cost trellis for a given cost framework, generalizing to MFC codes the trellis construction method of [5, 6]. Finally, Section VI gives some concluding remarks, including a brief discussion of some of the applications of these results. II Minimal Trellises A Terminology Throughout this paper, for integers a and b, we let [a; b] denote the set fa; a 1; bg, which is taken to be empty if a ....

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J. K. Wolf, "Efficient maximum-likelihood decoding of linear block codes using a trellis," IEEE Trans. on Inform. Theory, vol. IT-24, pp. 76--80, 1978.

.... Introduction of the trellis in coding theory traces back to the work of Forney [3] see also [4] The connection between finite state Markov chains and trellises is a natural one, and in this context trellises for binary linear block codes were mentioned in the work of Bahl, et al. 5] Wolf [6] used the parity check based construction of Bahl, et al. to construct block code trellises and proposed use of the Viterbi algorithm for soft decision decoding. At about the same time, Massey introduced an alternative construction procedure [1] and pointed out that re arranging the order of the ....

....with a single bit; the vertices through which the all zeros codeword pass would be labeled with a 0 and the other states with a 1. Note that this vertex label (or choice of state) encodes which rows of the generator matrix are active. By contrast, the vertex label in Wolf s construction [6] encodes a partial syndrome, and so is based on the parity check matrix. When the given generator matrix is trellis oriented both approaches yield essentially the same minimal trellis. 4 Minimal Trellises for Linear Codes In the previous section we showed how to construct a trellis from the ....

[Article contains additional citation context not shown here]

J. K. Wolf, "Efficient maximum-likelihood decoding of linear block codes using a trellis," IEEE Trans. on Inform. Theory, vol. IT-24, pp. 76--80, 1978.

....is investigated based on the input redundancy weight enumerating function using an uniform interleaver . An uniform interleaver is defined as a probabilistic device which maps a given input word of weight w into all distinct permutations # k w # with equal probability # k w # #1 . In [64] it was shown that soft decision maximum likelihood decoding of any (n,k) linear block code over GF(q) can be accomplished using the Viterbi algorithm applied to a trellis with no more than q (n#k) states. The trellis starts and ends in the same zero state. This characteristic allows the use of ....

....decoding algorithm without the need of appending any tail to the information sequence in order to drive the encoder to the zero state. 118 6. 3 Turbo coding the (15, 11) binary cyclic code Here we consider the same (15, 11) binary cyclic code with generator polynomial g(x) x 4 x 1 as in [64]. This is a Hamming code having a minimum Hamming distance of 3, so it is a single error correcting code. The encoder design is given in Figure 6.2. Initially the switch is in position 1 and the gate is open. The first eleven information bits are entered in the shift register and at the same time ....

J. K. Wolf, "Efficient maximum likelihood decoding of linear block codes using a trellis," IEEE Trans., vol. IT--24, pp. 76--80, Jan. 1978. 145

....code is NP complete. Keywords: permutation trellis complexity, NP completeness. I. Introduction Although the codes obtained by permuting the coordinates of a linear block code are equivalent, it is well known that the minimal trellises for these equivalent codes in general are not equivalent [1 7]; in particular, different coordinate permutations may yield trellises with different state complexity profiles. As J. L. Massey pointed out some time ago [2] the art of trellis decoding of a linear code [is] that of re arranging the order of digits in the code word to obtain a non systematic ....

....vertex from the root as the depth or time index of that particular vertex. Vertices with the same depth are referred to as the trellis states at that depth. The minimal trellis for a linear block code C over GF (q) can be constructed in a number of different (but equivalent) ways, e.g. [1, 2, 12]. For linear codes, the set of states in the minimal trellis at a given depth i can be characterized [3, 18] as a vector space over GF (q) Denoting the dimension of this vector space by s i (C) this means that there are q s i (C) states at depth i. The vector (s 0 (C) s n (C) is the ....

J. K. Wolf, "Efficient maximum-likelihood decoding of linear block codes using a trellis," IEEE Trans. on Inform. Theory, vol. IT-24, pp. 76--80, 1978.

....We have applied this algorithm to various codes, and our results are tabulated. 1 Introduction Although the codes obtained by permuting the coordinates of a linear block code are equivalent, it is well known that the minimal trellises for these equivalent codes in general are not equivalent [1, 2, 3, 4, 5, 6, 7]; in particular, different coordinate permutations may yield trellises with different state complexity profiles. As J. L. Massey pointed out some time ago [2] the art of trellis decoding of a linear code [is] that of re arranging the order of digits in the code word to obtain a non systematic ....

....reach a given vertex from the root as the time index of that particular vertex. Vertices with the same time index are referred to as the states at that time index. The minimal trellis for a linear block code C over GF (q) can be constructed in a number of different (but equivalent) ways, e.g. [1, 2, 11]. For linear codes, the set of states in the minimal trellis at a given time index i can be characterized [3, 12] as a vector space over GF (q) Denoting the dimension of this vector space by s i (C) this means that there are q s i (C) states at time index i. The vector (s 0 (C) s n ....

J. K. Wolf, "Efficient maximum-likelihood decoding of linear block codes using a trellis," IEEE Trans. on Inform. Theory, vol. IT-24, pp. 76--80, 1978.

....tight, the result of Theorem 3.41 ensures simpler decoding than the syndrome trellis algorithm for all linear codes and all code rates R; 0 R 1. 3.4.3 Notes Channels and maximum likelihood decoding are studied in Gallager [70] 3.4.1. Trellis decoding was introduced in Bahl et al. 18] Wolf [166] (see Chapter xx (Vardy) for the history and more results) Algorithm 3.8 is known in coding theory as the Viterbi algorithm and is very close to Dijkstra s shortest 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 a a b c d e Figure 3.3: Complexity of soft decision decoding algorithms for binary codes ....

J. K. Wolf, "Efficient maximum likelihood decoding of linear codes using a trellis, " IEEE Trans. Inform. Theory, 24 (1) (1978), 76--80.

....07974 Abstract It was conjectured by McEliece that among all trellises representing a binary linear block code the Wolf trellis minimizes the number of metric comparisons required by the Viterbi algorithm. In this short paper we give a proof of this conjecture. 1 Introduction In his 1978 paper [1], Wolf gave a compact representation of binary linear block codes by means of a trellis derived from the parity check matrix of the code. This paper motivated many researchers to find efficient trellis representations of block codes, see [2] for a list of references) Although there is no ....

J. K. Wolf, "Efficient maximum likelihood decoding of linear block codes," IEEE Trans. Inform. Theory, vol. IT--24, pp. 76--80, 1978.

No context found.

J.K. Wolf, "Efficient maximum-likelihood decoding of linear block codes," IEEE Trans. Inform. Theory, vol. 24, pp. 76--80, 1978.

No context found.

J.K. Wolf, "Efficient Maximum Likelihood Decoding of Linear Block Codes Using a Trellis," IEEE Tansactions on Information Theory, Vol. IT-24, No.1, pp. 76-80, Jan. 1986.

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