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Spacetime codes for high data rate wireless communication: Performance criterion and code construction
 IEEE Trans. Inform. Theory
, 1998
"... Abstract — We consider the design of channel codes for improving the data rate and/or the reliability of communications over fading channels using multiple transmit antennas. Data is encoded by a channel code and the encoded data is split into � streams that are simultaneously transmitted using � tr ..."
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Cited by 1225 (25 self)
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Abstract — We consider the design of channel codes for improving the data rate and/or the reliability of communications over fading channels using multiple transmit antennas. Data is encoded by a channel code and the encoded data is split into � streams that are simultaneously transmitted using � transmit antennas. The received signal at each receive antenna is a linear superposition of the � transmitted signals perturbed by noise. We derive performance criteria for designing such codes under the assumption that the fading is slow and frequency nonselective. Performance is shown to be determined by matrices constructed from pairs of distinct code sequences. The minimum rank among these matrices quantifies the diversity gain, while the minimum determinant of these matrices quantifies the coding gain. The results are then extended to fast fading channels. The design criteria are used to design trellis codes for high data rate wireless communication. The encoding/decoding complexity of these codes is comparable to trellis codes employed in practice over Gaussian channels. The codes constructed here provide the best tradeoff between data rate, diversity advantage, and trellis complexity. Simulation results are provided for 4 and 8 PSK signal sets with data rates of 2 and 3 bits/symbol, demonstrating excellent performance that is within 2–3 dB of the outage capacity for these channels using only 64 state encoders.
On the Trellis Structure of Block Codes
, 1995
"... The problem of minimizing the vertex count at a given time index in the trellis for a general (nonlinear) code is shown to be NPcomplete. Examples are provided that show that 1) the minimal trellis for a nonlinear code may not be observable, i.e., some codewords may be represented by more than one p ..."
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Cited by 56 (7 self)
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The problem of minimizing the vertex count at a given time index in the trellis for a general (nonlinear) code is shown to be NPcomplete. Examples are provided that show that 1) the minimal trellis for a nonlinear code may not be observable, i.e., some codewords may be represented by more than one path through the trellis and 2) minimizing the vertex count at one time index may be incompatible with minimizing the vertex count at another time index. A trellis product is defined and used to construct trellises for sum codes. Minimal trellises for linear codes are obtained by forming the product of elementary trellises corresponding to the onedimensional subcodes generated by atomic codewords. The structure of the resulting trellis is determined solely by the spans of the atomic codewords. A correspondence between minimal linear block code trellises and configurations of nonattacking rooks on a triangular chess board is established and used to show that the number of distinct minimal li...
Algorithmic Complexity in Coding Theory and the Minimum Distance Problem
, 1997
"... We start with an overview of algorithmiccomplexity problems in coding theory We then show that the problem of computing the minimum distance of a binary linear code is NPhard, and the corresponding decision problem is NPcomplete. This constitutes a proof of the conjecture Bedekamp, McEliece, van T ..."
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Cited by 34 (2 self)
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We start with an overview of algorithmiccomplexity problems in coding theory We then show that the problem of computing the minimum distance of a binary linear code is NPhard, and the corresponding decision problem is NPcomplete. This constitutes a proof of the conjecture Bedekamp, McEliece, van Tilborg, dating back to 1978. Extensions and applications of this result to other problems in coding theory are discussed.
SoftDecision Decoding of ReedMuller Codes as Generalized Multiple Concatenated Codes
 IEEE Trans. Inform. Theory
, 1995
"... In this paper, we present a new softdecision decoding algorithm for ReedMuller codes. It is based on the GMC decoding algorithm proposed by Schnabl and Bossert [1] which interprets ReedMuller codes as generalized multiple concatenated codes. We extend the GMC algorithm to listdecoding (LGMC). A ..."
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Cited by 13 (1 self)
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In this paper, we present a new softdecision decoding algorithm for ReedMuller codes. It is based on the GMC decoding algorithm proposed by Schnabl and Bossert [1] which interprets ReedMuller codes as generalized multiple concatenated codes. We extend the GMC algorithm to listdecoding (LGMC). As a result, a SDML decoding algorithm for the first order ReedMuller codes is obtained. Moreover, the performance achieved with LGMC for ReedMuller codes of higher order is considerably better compared to GMC. In particular, for the ReedMuller codes of length ¢¡¤ £ , quasi SDML decoding performance is obtained at a computational complexity that is by far less than optimum decoding using the syndrome trellis [2]. Simulations will also show that for ReedMuller codes up to a length 1024, the performance of LGMC decoding is more than 1dB superior to conventional GMC decoding. 1
Good Trellises for IC Implementation of Viterbi Decoders for Linear Block Codes
, 1996
"... This paper investigates trellis structures of linear block codes for the IC (integrated circuit) implementation of Viterbi decoders capable of achieving high decoding speed while satisfying a constraint on the structural complexity of the trellis in terms of the maximum number of states at any parti ..."
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Cited by 3 (0 self)
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This paper investigates trellis structures of linear block codes for the IC (integrated circuit) implementation of Viterbi decoders capable of achieving high decoding speed while satisfying a constraint on the structural complexity of the trellis in terms of the maximum number of states at any particular depth. Only uniform sectionalizations of the code trellis diagram are considered. An upper bound on the number of parallel and structurally identical (or isomorphic) subtrellises in a proper trellis for a code without exceeding the maximum state complexity of the minimal trellis of the code is first derived. Parallel structures of trellises with various section lengths for binary BCH and ReedMuller (R.M) codes of lengths 32 and 64 are analyzed. Next, the complexity of IC implementation of a Viterbi decoder based on an Lsection trellis diagram for a code is investigated. A structural property of a Viterbi decoder called ACSconnectivity which is related to state connectivity is introduced. This parameter affects the complexity of wirerouting (interconnections within the IC). The effect of five parameters namely: (1) effective computational complexity; (2) complexity of the ACScircuit; (3) traceback complexity; (4) ACSconnectivity; and (5)
Computational Methods in Coding Theory
, 1996
"... We consider various computational techniques in algebraic coding theory along two lines of work. First we investigate optimization of nonlinear codes by relaxing minimum distance constraints, developing, in the process, two algorithms for improving a given nonlinear code and a method of visualizin ..."
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Cited by 3 (1 self)
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We consider various computational techniques in algebraic coding theory along two lines of work. First we investigate optimization of nonlinear codes by relaxing minimum distance constraints, developing, in the process, two algorithms for improving a given nonlinear code and a method of visualizing algebraic codes in three dimensions. Secondly, we study the Generalized Lexicographic Construction, and show that it produces as special cases the lexicodes and derivatives with properties such as trellisorientation, trellisstate boundedness, and local optimality. We implement algorithms for generating these families of codes and, in the process, improve upon work by Conway and Sloane, Brualdi and Pless, Kschischang and Horn, and Zhang.
Ordered Binary Decision Diagrams and Minimal Trellises
 IEEE Transactions on Computers
, 1999
"... Ordered binary decision diagrams (OBDDs) are graphbased data structures for representing Boolean functions. They have found widespread use in computeraided design and in formal verification of digital circuits. Minimal trellises are graphical representations of errorcorrecting codes that play a p ..."
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Cited by 2 (1 self)
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Ordered binary decision diagrams (OBDDs) are graphbased data structures for representing Boolean functions. They have found widespread use in computeraided design and in formal verification of digital circuits. Minimal trellises are graphical representations of errorcorrecting codes that play a prominent role in coding theory. This paper establishes a close connection between these two graphical models, as follows. Let C be a binary code of length n, and let f C (x 1 ; : : : ; x n ) be the Boolean function that takes the value 0 at x 1 ; : : : ; x n if and only if (x 1 ; : : : ; x n ) 2 C . Given this natural oneto one correspondence between Boolean functions and binary codes, we prove that the minimal proper trellis for a code C with minimum distance d ? 1 is isomorphic to the singleterminal OBDD for its Boolean indicator function f C (x 1 ; : : : ; x n ). Prior to this result, the extensive research during the past decade on binary decision diagrams  in computer engineering ...
Introducton to the Special Issue on Codes and Complexity
, 1996
"... This paper continues the work of Lafourcade and Vardy [18], tabulated on ..."
CONSTRAINT COMPLEXITY OF REALIZATIONS OF LINEAR CODES ON ARBITRARY GRAPHS
, 805
"... ABSTRACT. A graphical realization of a linear code C consists of an assignment of the coordinates of C to the vertices of a graph, along with a specification of linear state spaces and linear “local constraint ” codes to be associated with the edges and vertices, respectively, of the graph. The κco ..."
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ABSTRACT. A graphical realization of a linear code C consists of an assignment of the coordinates of C to the vertices of a graph, along with a specification of linear state spaces and linear “local constraint ” codes to be associated with the edges and vertices, respectively, of the graph. The κcomplexity of a graphical realization is defined to be the largest dimension of any of its local constraint codes. κcomplexity is a reasonable measure of the computational complexity of a sumproduct decoding algorithm specified by a graphical realization. The main focus of this paper is on the following problem: given a linear code C and a graph G, how small can the κcomplexity of a realization of C on G be? As useful tools for attacking this problem, we introduce the VertexCut Bound, and the notion of “vctreewidth ” for a graph, which is closely related to the wellknown graphtheoretic notion of treewidth. Using these tools, we derive tight lower bounds on the κcomplexity of any realization of C on G. Our bounds enable us to conclude that good errorcorrecting codes can have lowcomplexity realizations only on graphs with large vctreewidth. Along the way, we also prove the interesting result that the ratio of the κcomplexity of the best conventional trellis realization of a lengthn code C to the κcomplexity of the best cyclefree realization of C grows at most logarithmically with codelength n. Such a logarithmic growth rate is, in fact, achievable. 1.