R. Michael Tanner. A recursive approach to low complexity codes. IEEE Trans. on Information Theory, Vol. 27, No. 5, pp. 533--547, September, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....[4] They have again become interesting because of the success of iterative decoding for Turbo codes. LDPC codes are competitors of these codes in performance of iterative decoding algorithms, as their performance approaches the Shannon limit [9] Tanner s graphical representation of LDPC codes [10] influenced much of the current literature. Most of these codes are constructed randomly, but explicit constructions are needed for implementation purposes as well as for knowing the properties of these codes. We give such constructions based on constructions of graphs with good girth. Let m 2 ....

....q) be the incidence matrix of lines and points of D(m, q) where rows are indexed by lines and columns are indexed by points, and consider it and its transpose to be parity check matrices of binary codes of length q called LU(m, q) codes. In other words, we take D(m, q) to be the Tanner graph [10] of the LDPC code LU(m, q) and investigate the properties of these codes. As the rows as well as the columns of H(m, q) are linearly dependent, the dimensions of these codes need to be determined. In [7] it is shown that (#) any two rows (columns) of H(m, q) have a 1 in at most one common column ....

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R. M. Tanner, "A recursive approach to low complexity codes," IEEE Trans. Inform. Theory, vol. IT - 27, pp. 533--547, 1981.

....imposed on H p . Our goal is to develop encoding techniques which converge quickly and which re use the sum product message passing decoder architecture described in [7] The idea behind using the code constraints to perform encoding on the graph is not new and was originally suggested by Tanner [8]. The work presented here forms a link between this concept and classical iterative matrix inversion techniques, allowing the design of good codes that encode quickly. By reusing the decoder architecture for encoding, both operations can be performed by the same circuit on a time switched basis. ....

R. M. Tanner, "A recursive approach to low complexity codes," IEEE Trans. Inform. Theory, vol. 27, pp. 533--547, Sep. 1981.

....4 establishes that the LP decoder has a performance guarantee specified by the fractional distance of the code. B. Lower Bound on the Fractional Distance The following theorem asserts that the fractional distance is exponential in the girth of G. It is analogous to an earlier result of Tanner [Tan81] which provides a similar bound on the classical distance of a code in terms of the girth of the associated Tanner graph. Theorem 5 Let G be a Tanner graph with variable degree d # # 3 and check degree dr # 2, and let g be the girth of G. Then the fractional distance is at least (2 dr ) d ....

R. M. Tanner. A recursive approach to low complexity codes. Information Theory, IEEE Transactions on, 27(5):533-- 547, 1981.

....I connectivity II,III c o n n e c t i v i t y Fig. 4. Depending on the connectivity the general decoder building block implements boxplus or an arithmetic summation of # values. have L c # ###### # P # #=P # #: 13) Analog Decoders Based on Graphs, LDPC Decoders: With Tanner graphs [9]we have a graphical representation of the check structure of channel codes. Here the function nodes are called check nodes and are representations of valid parity check equation for the particular code. In addition we have the summing functions in the variable node. On the receive side we ....

R. Tanner, "A recursive approach to low complexity codes," IEEE Trans. Inform. Theory, vol. IT-27, pp. 533--547, Sept. 1981.

....e ective pseudocodeword weight can be less than the minimum codeword weight. Key words. Codes on graphs, iterative decoding, pseudocodewords, e ective weights, tail biting. AMS(MOS) subject classi cations. 94B99 1. Introduction. The subject of codes de ned on graphs was founded by Tanner [10], inspired by Gallager s low density parity check (LDPC) codes [4] The thesis of Wiberg [12, 13] along with the practical successes of turbo codes and LDPC codes, has stimulated great current interest in this subject. For recent developments, see [1, 6, 7, 8] By now it is well known that if C ....

R. M. Tanner, \A recursive approach to low complexity codes," IEEE Trans. Inform. Theory, vol. IT-27, pp. 533-547, Sept. 1981.

....has no cycles. It has been recognized that the message passing algorithm is an instance of Pearl s belief propagation which converges to the optimal solution if the operating graph is cycle free. The basic idea of probability inference decoding is implied in Tanner s pioneering work in 1981 [18], and later studied by Wiberg [19] Frey [20] Forney et al. [21] as it gained success in decoding LDPC codes. However, little has been reported for practical application on convolutional codes. This is because the code graph of a convolutional code is in general complex and involves many cycles ....

....between the two approaches is about 0.1 dB at bit error rate of 10 5 . Hence, parallel sum product decoding serves as a good candidate for hardware implementation. 8 Comparison to Other Related Codes Graphical representation of codes has shed great insight into the understanding of many codes [18] [19] 20] 21] including turbo codes, LDPC codes and (irregular) repeat accumulate (RA IRA) codes [15] 17] This section revisits PA codes from the graph perspective for a comparison and unification of PA codes and other capacity approaching codes. The Tanner graph structure shown in Fig. 19 ....

R. M. Tanner, "A recursive approach to low complexity codes," IEEE Trans. Inform. Theory, vol. IT-27, pp.533-547, Sept. 1981.

....close tocapacity performance shifted emphasis to iterative decoding techniques. One particular class of codes that can be iteratively decoded is that of expander codes. An expander code is constructed by assigning binary digits to edges of a bipartite graph in a way that was introduced by Tanner [14]. A surge of interest in them occurred after it was shown in [13] that if the underlying graph is an expander graph, then they correct an N) number of errors under an O(N) iterative decoding. Not only was this a signi cant achievement for iterative decoding but it was the rst example of this ....

M. Tanner, \A recursive approach to low-complexity codes," IEEE Trans. Inform. Theory, IT-27, no 5, pp.533-547, 1981.

....work is that while capacity has been shown to be effectively achieved for long codes, the relative performance of coding schemes for shorter block lengths is not well understood. Our approach is inspired by Tanner s formulation of codes defined hierar chically on algebraically constructed graphs [11], and by Sipset and Spielman s analysis [10] showing the im portance of graph expansion. In this paper we study three families of explicit expander codes. The first family of codes is closely related to the explicit asymptotically good family of ex pander codes constructed in [10] These codes ....

....algorithm makes is invalid after a number of iterations equal to the girth of the graph. Taking p 13, we obtain 14 regular graphs on the group G PSL2 (Fq) with q(q2 1) 2 vertices. To con 1 struct a code with rate on these 14 regular graphs, we use the following procedure By puncturing the [15, 11, 3] Hamming code we obtain a [14, 11] code . Expurgating the even codewords of q yields a [14, 10] code q2. Assigning q to a proportion 0 a 1 of the constraint nodes, and q2 to the remaining proportion of 1 a constraint nodes, we obtain a code with overall rate of at least 1 3a 4(1 a) 3 a ....

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R. M. Tanner, "A recursive approach to low com- plexity codes," IEEE Trans. on Information Theory, Vol. 27, No. 5, September, 1981, pp. 533 547.

....based on factor graphs with binary nodes Similar to the CID algorithm the first concept of our analog decoder is based on the factor graph with binary variable nodes. If the factor graph is derived from the H matrix of the code, this graphical representation is equivalent to the Tanner graph [13]. Here the variable (or symbol) nodes correspond to the code symbols, which are connected to function (or check) nodes corresponding to the parity check equations of the code. Each function node is then a representation of a parity check equation for the particular code, and the set of variables ....

....block codes, e.g. the (7,4) Hamming code with the parity check matrix H (1 1 0 1 1 0 0) I 0 0 0 , 14 0 i i i 0 0 1 or for the analog decoding of LDPC codes. Of great interest are the original LDPC codes found by Cal laget [10] as well as modified codes such as the generalized LDPC codes [13], 14] or LDPC codes based on finite geometries [15] A selection of those codes with short block length can be found in Table 2. In the decoder network, which operates with Lvalues, the parity check operation in the function nodes will be converted into the boxplus [ operation and at variable ....

R. Tanner, "A recursive approach to low complex- ity codes," IEEE Trans. Inform. Theory, vol. IT-27, pp. 533-547, Sept. 1981.

....coding theory literature. Gallager s low density parity check (LDPC) codes [1] were the earliest appearance of iteratively decodable codes and much of modern coding theory could have been derived in the early 1960s. Other notable contributions to the area of codes on graphs are the work of Tanner [2], Wiberg et al. 3, 4] and a number of papers exploring the interplay between iterative decoding and other graph based algorithms [5, 6, 7, 8, 9] Trellis representations of codes were introduced by Forney [10] in order to explain the workings of the Viterbi algorithm. However it soon became ....

R. M. Tanner, \A recursive approach to low complexity codes," IEEE Trans. Inform. Theory, vol. IT-27, pp. 533-547, Sept. 1981.

....introduced in [14] Many ideas used in the analysis in [11] 12] go back to the original work by Gallager [13] who used tree like graphs to describe his probabilistic decoding algorithm for low density parity check codes. Gallager s algorithm was later extended to a wider range of codes by Tanner [16]. A description of turbo codes by a Tanner graph is given in [17] In this paper we examine a family of parallel concatenated codes, the superorthogonal turbo codes [1] based on superorthogonal convolutional codes [2] 3] 18] Iterative decoding of these codes is shown to result in very good ....

R. M. Tanner, \A recursive approach to low complexity codes," IEEE Trans. on Inform. Theory, vol. IT-27, pp. 533-547, September 1981.

....and decoding algorithm. The supremum of over all (J; K) codes and decoding algorithms we will call iterative low density limit and designate as p 0 = p 0 (J; K) In this paper we derive a lower bound p 0 on p 0 . Low density codes can be described both by bipartite graphs due to Tanner [5] as well as by tree like Gallager graphs, as introduced in the next paragraph. x 2 Graph representation of low density codes For any code symbol v n , n = 0; 1; N 1, we can build a tree like graph. To describe this graph we introduce a special terminology. Each node within an even level ....

R. M. Tanner, \A Recursive Approach to Low Complexity Codes", IEEE Trans. Inform. Theory, vol. 27, no. 9, pp. 533-547, Sep 1981.

....using special constructive methods. From our point of view this way is preferable for the analysis of iterative decoding of codes. The two approaches were introduced in the work of Gallager [2] and developed, particularly, in the papers of Zyablov and Pinsker [3] Margulls [4] 5] 6] Tanner [7], Benedetto and Montorsi [8] and other recently published works. 1This work was supported in part by Swedish Research Council for Engineering Sciences (Grant 98 216) In this article we adhere to the second approach and analyze the asymptotic properties of iterative decoding of binary turbo ....

Tanner R. M., A Recursive Approach to Low Complexity Codes. // IEEE Trans. Inform. Theory. 1981. V. 27, N. 9, P. 533 547.

....University Jerusalem 91904 Israel, Email: johnblue cs.huji.ac.il y Institute of Computer Science, Hebrew University Jerusalem 91904 Israel, Email: shlomoh cs.huji.ac.il 1 1 Introduction We propose a new family of asymptotically good binary codes. Our work is motivated by earlier constructions [4, 3, 6]. We use standard terms from coding theory. See [5] for a general reference to this area. In [4] Tanner suggested a method for constructing linear binary codes from a shorter code as follows: Let C 0 be a code of length , and G be a bipartite graph, where the vertices on the left side have ....

....University Jerusalem 91904 Israel, Email: shlomoh cs.huji.ac.il 1 1 Introduction We propose a new family of asymptotically good binary codes. Our work is motivated by earlier constructions [4, 3, 6] We use standard terms from coding theory. See [5] for a general reference to this area. In [4], Tanner suggested a method for constructing linear binary codes from a shorter code as follows: Let C 0 be a code of length , and G be a bipartite graph, where the vertices on the left side have degree . Denote the number of vertices on the right side by n. The resulting code C has length n, ....

M. Tanner. A recursive approach to low-complexity codes. IEEE Trans. Inform. Theory, IT-27:533547, 1981.

....Our analysis of iterative decoding is based on the representation of a LDC code as a 1 The material in this paper was presented in part at the Sixth International Workshop on Algebraic and Combinatorial Coding Theory, Pskov, Russia, September 6 12 1998. 1 graph, analogously to Tanner [8]. In the end of the paper, simulation results are given for iterative decoding of homogeneous LDC codes. In the second paper a special statistical ensemble of LDC codes, described by Markov equations, will be studied. For these codes a lower bound on the free distance, and upper bounds for ....

R. M. Tanner, "A Recursive Approach to Low Complexity Codes", IEEE Transactions on Information Theory, vol. IT-27, pp. 533-547, Sept. 1981.

....years when applied to turbo codes [5] Since the complexity of iterative decoding practically does not depend on the memory of the code, it can be used for decoding very long codes, and very good performance can be achieved even for transmission close to the Shannon limit. Using a graph approach [1,7], we describe and analyze an iterative algorithm for decoding of homogeneous LDPC convolutional codes. For simplicity we consider only the case when the code rate R = 1 2. In the end we present the results of computer simulations of the algorithm for di#erent LDPC convolutional codes, and compare ....

R.M. Tanner, "A Recursive Approach to Low Complexity Codes", IEEE Transactions on Information Theory, vol. IT-27, pp. 533-547, Sept. 1981.

....0 0 1 1 1 0 0 1 # # # # # # # Figure 1: The vector (0, 0, 0, 1, 0, 0, 1, 0) gives rise to four violated constraints (striped) and thus is not a codeword. The right hand side shows the parity check matrix corresponding to the graph. ing the girth of the graphs that Gallager [2] and Tanner [11] made was insu#cient to show the existence of asymptotically good families of low density codes. Definition 1 A graph # = V, E) with n vertices is said to be an # expander if for any vertex subset S # V with S # n 2 , #(S) # # S , where #(S) # t : s # S, t ## S, ....

R. Michael Tanner. A recursive approach to low complexity codes. IEEE Trans. on Information Theory, IT27 (5):533--547, September, 1981.

.... channel capacity has recently been numerically demonstrated for various memoryless [9] 10] channels using Gallager codes, also known as low density parity check (LDPC) codes [11] The theory of Gallager codes has vastly bene tted from the notion of codes on graphs rst introduced by Tanner [12] and further expanded into a unifying theory of codes on graphs by Wiberg et.al. 13] and Forney [14] MacKay [15] 16] showed that there exist Gallager codes that outperform turbo codes [17] A major breakthrough was the construction of irregular Gallager codes [18] and the development of a ....

....and denotes binary vector addition. The codeword s satis es H s = c = c 1 ; c 2 ; c n k ] T = H r: 7) The code is linear if and only if c = 0, otherwise, the code is a coset code of a linear Gallager code. It is convenient to represent a Gallager coset code by a bipartite graph [12], 14] 23] The graph has two types of nodes: n variable nodes (one variable node for each entry in the vector s) and n k check nodes (one check node for each entry in the vector c) There is an edge connecting the i th check node and the j th variable node if the entry H(i; j) in the i th row ....

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B. M. Tanner, \A recursive approach to low complexity codes," IEEE Trans. Inform. Theory, vol. 27, pp. 533{ 547, September 1981.

....is associated to the trivial parity check code (n; n 1) This representation of LDPC codes has been used by Sipser and Spielman [2] to study the in uence of the graph expansion on the code parameters. The graphical representation of block codes has been rst exploited and generalized by Tanner [3]. Tanner codes based on a bipartite deterministic graph are obtained by replacing the (n; n 1) code associated to one parity check node with a less trivial constituent code (n; k) Thus, building a Tanner code on a random graph (instead of a deterministic one) is a second method to construct GLD ....

R.M. Tanner: A recursive approach to low complexity codes, IEEE Trans. on Information Theory, Vol. IT27,

....complexity of a group code C is one of the main concerns in the application of C. It is believed that appropriate graphical representations of codes will contribute in this regard. The well known graphical models presented for linear codes are trellis diagram [1, 2, 3] Tanner graph (TG) [4], and Tanner Wiberg Loeliger graph [5, 6] While the Viterbi algorithm works with trellis diagrams, there is not any known algorithm working on Tanner graphs (TGs) with cycles. However, the forward backward algorithm [7] including the min sum and the sum product algorithms introduced in [4] work ....

....(TG) 4] and Tanner Wiberg Loeliger graph [5, 6] While the Viterbi algorithm works with trellis diagrams, there is not any known algorithm working on Tanner graphs (TGs) with cycles. However, the forward backward algorithm [7] including the min sum and the sum product algorithms introduced in [4], work on any cycle free TG, here called acyclic Tanner graph (ATG) A TG representing a linear block code with check matrix H = h ij ] is a bipartite graph in which one of the two sets of vertices denote the parity nodes, the rows of H, and the other set denote the symbol nodes, the columns of ....

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R. M. Tanner, "A recursive approach to low complexity codes," IEEE Trans. Infor. Theory, vol. IT-27, Sept. 1981, pp. 533-547.

....of Illinois at Urbana Champaign, Urbana, IL 61801 USA. A. Vardy is with the University of California at San Diego, La Jolla, CA 92093 0428 USA. Communicated by F. R. Kschischang, Associate Editor for Coding Theory. Publisher Item Identifier S 0018 9448(99)06101 5. It is proved in [2] 15] [26], and [30] that the min sum, the sum product, the GDL, and other versions of iterative decoding on factor graphs all converge to the optimal solution if the underlying factor graph is cycle free. If the underlying factor graph has cycles, very little is known regarding the convergence of iterative ....

....graph has cycles, very little is known regarding the convergence of iterative decoding methods. This work is concerned with an important special type of factor graphs, known as Tanner graphs. The subject dates back to the work of Gallager [11] on low density parity check codes in 1962. Tanner [26] extended the approach of Gallager [11] 12] to codes defined by general bipartite graphs, with the two types of vertices representing code symbols and checks (or constraints) respectively. He also introduced the min sum and the sum product algorithms, and proved that they converge on cycle free ....

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R. M. Tanner, "A recursive approach to low-complexity codes," IEEE Trans. Inform. Theory, vol. IT-27, pp. 533--547, 1981.

....(ML) decoding complexity of a group code C is one of the main concerns in the application of C. t is believed that appropriate graphical representations of codes will contribute in this regard. The well known graphical models presented for linear codes are trellis diagram [1 33, Tanner graph (TG) [4], and Tanner Wiberg Loeliger graph [5, 6] Bald [1] and Wolf [3] demonstrated that one could construct a trellis diagram for a block code, and hence employ the Viterbi algorithm [br its decoding. Wolf also gave a simple algorithm for the construction of the trellis diagram. Forney [2] later ....

....we conclude that all the base subcodes used in our discussions am indeed maximal. We will present an independent proof for the maximality of the extracted subcode for the case of the first order Reed Muller code. 2 Tanner graphs 2.1 Minimal Tanner graph Tanner graphs were first. introduced in [4]. A TO of a linear code C is a hipattire graph obtained from a set of parity check equations representing C. The two sets of vertices are called variable nodes and check nodes. A variable node v is adjacent with a check node u iff the coefficient of the variable corresponding with v is not zero in ....

TANNER, R.M.: 'A recursive approach to low complexity codes', IEEE Trans. [nf Theory, 1981, 1T-27, pp. 533-547

....field size, which contrasts with previous observations. Keywords LDPC codes, irregular and regular Tanner graphs, linear time encodable codes, girth IBM Research, Zurich Research Laboratory, 8803 Ruschlikon, Switzerland. Email: fxhu, ele, arng zurich.ibm.com I. INTRODUCTION ODES on graphs [1 12] have attracted considerable attention owing to their capacity approaching performance and low complexity iterative decoding. The prime examples of such codes are the lowdensity parity check (LDPC) codes. It is known that the belief propagation (BP) or sum product algorithm (SPA) over ....

....considerable attention owing to their capacity approaching performance and low complexity iterative decoding. The prime examples of such codes are the lowdensity parity check (LDPC) codes. It is known that the belief propagation (BP) or sum product algorithm (SPA) over cycle free Tanner graphs [1] provides optimum decoding, hence it is natural to try to minimize the influence of the cycles in the iterative decoding process. This approach has been adopted for both LDPC codes [13] and turbo codes [14] by using rather long block lengths. If the cycles are made long enough, the decoding ....

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R. M. Tanner, "A recursive approach to low complexity codes," IEEE Trans. Inform. Theory, vol. IT-27, pp. 533--547, Sept. 1981.

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R. Michael Tanner. A recursive approach to low complexity codes. IEEE Trans. on Information Theory, Vol. 27, No. 5, pp. 533--547, September, 1981.

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R. M. Tanner, "A recursive approach to low complexity codes," IEEE Trans. on Information Theory, Vol. 27, No. 5, September, 1981, pp. 533--547.

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R.M.Tanner, "A recursive approach to low-complexity codes," IEEE Trans. Inform. Theory, vol. 27, pp. 533--547, September 1981.

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R. M. Tanner, "A recursive approach to low complexity codes", IEEE Trans. on Inf. Th., vol. 27, no. 5, pp. 533547, Sept. 1981.

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R. M. Tanner, "A recursive approach to low complexity codes", IEEE Trans. on Inf. Th., vol. 27, no. 5, p. 533547, 1981.

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R. M. Tanner, "A recursive approach to low complexity codes," IEEE Trans. Inform. Theory, vol. IT-27, pp.533-547, Sept. 1981

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R. M. Tanner, "A recursive approach to low complexity codes," IEEE Trans. Inform. Theory, vol. IT-27, pp. 533--547, Sept. 1981.

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R. M. Tanner, "A recursive approach to low complexity codes," IEEE Trans. Inform. Theory, vol. IT - 27, pp. 533--547, 1981.

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R. M. Tanner. A recursive approach to low complexity codes. IEEE Transactions on Information Theory, 27(5):533-547, 1981.

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R.M. Tanner, "A recursive approach to low complexity codes," IEEE Transactions on Information Theory, vol. 27, September 1981.

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R. M. Tanner, "A recursive approach to low complexity codes," IEEE Trans. Inform. Theory, vol. IT-27, pp. 533--547, Sept. 1981.

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R. M. Tanner. A recursive approach to low complexity codes. IEEE Trans. Inform. Theory, 27:533--547, Sept. 1981.

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R. M. Tanner, "A recursive approach to low complexity codes," IEEE Trans. Inform. Theory, vol. IT-27, pp. 533--547, 1981.

No context found.

R. M. Tanner, "A recursive approach to low complexity codes," IEEE Trans. Information Theory, vol. 27, pp. 533--547, Sept. 1981.

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R. M. Tanner, "A recursive approach to low complexity codes," IEEE Trans. Inform. Theory, vol. 27, no. 5, pp. 533-547, Sep. 1981.

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R. M. Tanner, "A recursive approach to low-complexity codes," IEEE Trans. on Inform. Theory, vol. IT--27, pp. 533--547, Sept. 1981.

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R. M. Tanner, "A recursive approach to low complexity codes," IEEE Trans. Inform. Theory, vol. IT-27, no. 5, pp. 533--547, September 1981.

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R. M. Tanner, "A recursive approach to low complexity codes," IEEE Transactions on Information Theory, vol. IT-27, no. 9, pp. 533--547, Sept. 1981.

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R. M. Tanner. A recursive approach to low complexity codes. IEEE Transactions on Information Theory, 27(5):533-547, 1981.

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R. M. Tanner, "A recursive approach to low complexity codes," IEEE Trans. on Inform. Theory, vol. IT-27, pp. 533--547, Sept. 1981.

No context found.

R. M. Tanner. A recursive approach to low complexity codes. IEEE Trans. Inform. Theory, 27(5):533--547, Sept. 1981.

No context found.

R. Michael Tanner. A recursive approach to low complexity codes. IEEE Transactions on Information Theory, 27(5):533--547, 1981.

No context found.

R. M. Tanner, "A recursive approach to low complexity codes," IEEE Transactions on Information Theory, vol. 27, pp. 533--547, 1981.

No context found.

R. M. Tanner, "A recursive approach to low complexity codes," IEEE Trans. Inform. Theory, vol. IT-27, pp. 533--547, Sept. 1981.

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R. M. Tanner, "A recursive approach to low complexity codes," IEEE Trans. Inform. Theory, vol. IT-27, pp. 533--547, Sept. 1981.

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Tanner, R. M. A recursive approach to low complexity codes. IEEE Transactions on Information Theory 27, 5 (1981), 533--547.

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R. M. Tanner, "A Recursive Approach to Low Complexity Codes," IEEE Trans. Inform. Theory, vol. 27, no. 5, pp. 533-547, Sept. 1981.

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