J.L. Massey, "Foundation and methods of channel encoding," Proc. Int. Conf. Information Theory and Systems, vol. 65, pp. 148--157, NTG-Fachberichte, Berlin, 1978.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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 possible to ....

....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 regular trellis for a given block code, it was not until Forney s paper [47] that the construction of trellis diagrams for block codes truly gained research interest. The construction of a code trellis makes it ....

J. L. Massey, "Foundation and methods of channel encoding," Proceedings of the International Conference on Information Theory and Systems, NTG-Fachberichte, vol. 65, Berlin, September 1978.

....practical subclass of the linear cyclic codes, from which the combinatorial representation in the form of a graph can be obtained by standard procedures. Keywords: linear block codes, cyclic codes, Reed Solomon codes, tailbiting trellises. 1 Introduction Trellis descriptions of block codes [1,15,8,10,3,6,9] are combinatorial descriptions, as opposed to the traditional algebraic descriptions of block codes. A minimal conventional trellis for a linear block code is just the transition graph for the minimal finite state automaton which accepts the language consisting of the set of all codewords. With ....

J.L.Massey, Foundations and methods of channel encoding, in Proc. Int. Conf. on Information Theory and Systems 65(Berlin, Germany) Sept 1978.

....original automaton. Keywords: block codes, minimal trellis, decoder complexity, subtrellis overlaying. 1 Introduction The theory of finite state automata has many interesting connections to the field of error correcting codes [13] After the early work on trellis representations of block codes[1,22,15,18,7], there has recently been a spate of research on minimal trellis representations of block error correcting codes[8,12,17] Trellis descriptions are combinatorial descriptions, as opposed to the traditional algebraic descriptions of block codes. A minimal trellis for a linear block code is just the ....

....before decoding and are known to provide significant coding gains [4] when compared with hard decision decoding algorithms. That block codes have e#cient combinatorial descriptions in the form of trellises was discovered in 1974 [1] Other early seminal work in the area appears in [22] [15] [7] 18] For background on the algebraic theory of block codes, readers are referred to the classic texts [14, 2,16] for trellis structure of block codes, 19] is an excellent reference. Let F q be the field with q elements. It is customary to define linear codes algebraically as follows: ....

J.L.Massey, Foundations and methods of channel encoding, in Proc. Int. Conf. on Information Theory and Systems 65(Berlin, Germany) Sept 1978.

....of trellises was discovered in Presently at Nokia Research Center, Helsinki, Finland H. Reichel and S. Tison (Eds. STACS 2000, LNCS 1770, pp. 53 64, 2000. c Springer Verlag Berlin Heidelberg 2000 54 Kaustubh Deshmukh et al. 1974 [1] Two other early papers in this subject were [19] and [11]. A landmark paper by Forney [3] in 1988 began an active period of research on the trellis structure of block codes. It was realized that the well known Viterbi Algorithm [16] which is actually a dynamic programming shortest path algorithm) could be applied to soft decision decoding of block ....

J.L.Massey, Foundations and methods of channel encoding, in Proc. Int. Conf. on Information Theory and Systems 65(Berlin, Germany) Sept 1978.

....in the literature. For example, state complexity is called minimal trellis size in [2] absolute state complexity is called absolute minimal trellis size in [2] and minimal state complexity in [13] Finding a coordinate order of C that achieves s[C] is called the art of trellis decoding in [10] since exhaustive computation of s(C) over all possible coordinate orders of C is infeasible, even for quite short codes. An important step towards attaining this goal is determining good lower bounds on s[C] The dimension length pro le (DLP) of C is a deep property which is equivalent to the ....

J. L. Massey. Foundation and methods of channel encoding. In Proceedings of the International Conference on Information Theory and Systems, 1978.

....compared with hard decision decoding algorithms. That block codes have ecient combinatorial descriptions in the form of trellises was discovered in 1974 [1] The material in this paper was presented in part at STACS 2000, Lille, France 1 Two other early papers in this subject were [26] and [17]. A landmark paper by Forney [6] in 1988 began an active period of research on the trellis structure of block codes. It was realized that the well known Viterbi Algorithm [24] which is actually a dynamic programming shortest path algorithm) could be applied to soft decision decoding of block ....

J.L.Massey, Foundations and methods of channel encoding, in Proc. Int. Conf. on Information Theory and Systems 65(Berlin, Germany) Sept 1978.

....of a code, its size prevents it from being used in decoding all but the simplest of codes. Instead, for complicated codes, trellises [10, 13] are often used together with the Viterbi algorithm [33] 1.4 Trellis Decoding A trellis is a time indexed graph that represents a particular code. Massey [37] was the first to rigorously define the trellis, but the terminology of the definition has evolved since, and the following definition is loosely based on [33, p. 3] Definition 1.1 ( 37, 33] A trellis T = V; E) of rank n is a finite, directed, edgelabelled graph with the following ....

....[33] 1.4 Trellis Decoding A trellis is a time indexed graph that represents a particular code. Massey [37] was the first to rigorously define the trellis, but the terminology of the definition has evolved since, and the following definition is loosely based on [33, p. 3] Definition 1. 1 ([37, 33]) A trellis T = V; E) of rank n is a finite, directed, edgelabelled graph with the following properties: 5 1. each vertex v 2 V has an associated depth d 2 f1; 2; 3; ng 2. each edge e 2 E connects vertices at depth i to depth i 1, for some i 3. there is one vertex A at depth 0 and ....

J.L. Massey, "Foundation and methods of channel encoding," Proc. Int. Conf. Information Theory and Systems, NTG-Fachberichte, Berlin, 1978.

....when presented with an arbitrary point 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 . ....

J. L. Massey, Foundation and method of channel encoding, in Proc. Int. Conf. on Info. Theory and Systems, NTG-Fachberichte (Berlin), 65 (1978), 148-157.

....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 Viterbi algorithm explores all states in the ....

J. L. Massey, "Foundation and methods of channel encoding," in Proc. Int. Conf. Inform. Theory and Systems, vol. 65, (Berlin), Sept. 1978.

....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 and lattices, have a well defined trellis structure. Forney s paper sparked a flurry of activity in this area ....

....by y in C, or y follows x in C. For sets of label strings X and Y , let XY = fxy : x 2 X; y 2 Y g. The set XY is said to have product form. B Block Trellises The following definition generalizes the definition of a trellis given in [1] which in turn was based on the trellis definitions of [7, 10]. Definition. A trellis is an edge labeled digraph T = V; E; A) with the property that, given any two vertices v; v 0 2 V , if there is a path of length n from v to v 0 in T , then every path from v to v 0 in T has length n. A sub trellis is a subgraph of a trellis. In a (block) trellis, ....

J. L. Massey, "Foundation and methods of channel encoding," in Proc. Int. Conf. Inform. Theory and Systems, vol. 65, (Berlin), Sept. 1978.

....2. the goal can be reached from all vertices; 3. the number of edges traversed in passing from the root to the goal along any path is n; and 4. the set of n tuples obtained by reading off the edge labels encountered in traversing all paths from the root to the goal is C. This definition follows [1, 2] but differs slightly in that all edges in the graph must be labeled and the tree associated with a code is not a trellis. It is useful to regard the number of edges needed to traverse from the root to a given vertex as the time index associated with that vertex. Block code trellises are of both ....

....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 codeword digits can sometimes lead to smaller, simpler trellises. Interest in the trellis structure of block codes appeared dormant until the appendix of Forney s coset codes paper [7] sparked a flurry of activity in this area [2, 8, 9, 10, 11, ....

J. L. Massey, "Foundation and methods of channel encoding," in Proc. Int. Conf. Inform. Theory and Systems, vol. 65, (Berlin), Sept. 1978.

....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 code for which [the trellis complexity] is as small as possible. In this paper we show that the problem of minimizing the ith component of the state complexity ....

....Max Cut [8,16] establishes the NP completeness of the graph theoretic PRP problem, which, by restriction, establishes the NP completeness of i PRP. Some concluding remarks are made in Section V. II. State Complexity Profiles Formal definitions of block code trellises are given, e.g. in [2, 12, 17]. Briefly, a trellis for a block code C of length n is an edge labeled directed graph with a distinguished root vertex having in degree zero and a distinguished goal vertex have out degree zero, with the properties that: 1. all vertices can be reached from the root; 2. the goal can be reached ....

[Article contains additional citation context not shown here]

J. L. Massey, "Foundation and methods of channel encoding," in Proc. Int. Conf. Inform. Theory and Systems, vol. 65, (Berlin), Sept. 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 ....

....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 code for which [the trellis complexity] is as small as possible. In this paper we show that a related problem, that of minimizing the ith component of the ....

[Article contains additional citation context not shown here]

J. L. Massey, "Foundation and methods of channel encoding," in Proc. Int. Conf. Inform. Theory and Systems, vol. 65, (Berlin), Sept. 1978.

....We now realize that these are two instances of the same result, described in different languages. A central problem in the study of OBDDs is how to order the variables for a given function so that the size of the resulting decision diagram is minimized. A similar problem for trellises, known [61, 75] as the art of trellis decoding or the permutation problem, asks how the time axis for a given code should be permuted in order to minimize the complexity of the resulting trellis. Once again, these are essentially two instances of the same problem. In both cases, the research is centered around ....

J.L. Massey, "Foundation and methods of channel encoding," Proc. Int. Conf. Information Theory and Systems, vol. 65, pp. 148--157, NTG-Fachberichte, Berlin, 1978.

No context found.

J.L. Massey, "Foundation and methods of channel encoding," Proc. Int. Conf. Information Theory and Systems, vol. 65, pp. 148--157, NTG-Fachberichte, Berlin, 1978.

No context found.

J. L. Massey, "Foundation and methods of channel encoding," in Proc. Int. Conf. Information Theory and Systems, Berlin, Germany, Sept. 1978.

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