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1.
Let a q-ary linear (n, k) code C be used over a memoryless channel. We design a decoding algorithm ΨN that splits the received block into two halves in n different ways. First, about √N error patterns are found on either half. Then the left- and right-hand lists are sorted out and matched to form codewords. Finally, the most probable codeword is chosen among at most n√N codewords obtained in all n trials. The algorithm can be applied to any linear code C and has complexity order of n3√N. For any N⩾qn-k, the decoding error probability PN exceeds at most 1+qn-k/N times the probability PΨ (C) of maximum-likelihood decoding. For code rates R⩾1/2, the complexity order qn-k/2 grows as square root of general trellis complexity qmin{n-k,k}. When used on quantized additive white Gaussian noise (AWGN) channels, the algorithm ΨN can provide maximum-likelihood decoding for most binary linear codes even when N has an exponential order of qn-k  相似文献   

2.
Minimum distance decoding (MDD) for a general error-correcting linear code is a hard computational problem that recently has been shown to beNP-hard. The complexity of known decoding algorithms is determined bymin (2^{k},2^{n-k}), wherenis the code length andkis the number of information digits. Two new algorithms are suggested that reduce substantially the complexity of MDD. The algorithms use a new concept of zero neighbors--a special set of codewords. Only these codewords (which can be computed in advance) should be stored and used in the decoding procedure. The number of zero neighbors is shown to be very small compared withmin (2^{k},2^{n-k})forn gg 1and a wide range of code ratesR = k/n. For example, forR approx 0.5this number grows approximately as a square root of the number of codewords.  相似文献   

3.
Efficient maximum likelihood decoding of linear block codes using a trellis   总被引:4,自引:0,他引:4  
It is shown that soft decision maximum likelihood decoding of any(n,k)linear block code overGF(q)can be accomplished using the Viterbi algorithm applied to a trellis with no more thanq^{(n-k)}states. For cyclic codes, the trellis is periodic. When this technique is applied to the decoding of product codes, the number of states in the trellis can be much fewer thanq^{n-k}. For a binary(n,n - 1)single parity check code, the Viterbi algorithm is equivalent to the Wagner decoding algorithm.  相似文献   

4.
For a linear code C of length n and dimension k, Wolf (1978) noticed that the trellis state complexity s(C) of C is upper-bounded by w(C):=min(k,n-k). We point out some new lower bounds for s(C). In particular, if C is an algebraic-geometric code, then s(C)/spl ges/w(C)-(g-a), where g is the genus of the underlying curve and a is the abundance of the code.  相似文献   

5.
An iterative trellis search technique is described for the maximum-likelihood (ML) soft decision decoding of block codes. The proposed technique derives its motivation from the fact that a given block code may be a subcode for a parent code whose associated trellis has substantially fewer edges. Through the use of list-Viterbi (1967) decoding and an iterative algorithm, the proposed technique allows for the use of a trellis for the parent code in the ML decoding of the desired subcode. Complexity and performance analyses, as well as details of potential implementations, indicate a substantial reduction in decoding complexity for linear block codes of practical length while achieving ML or near-ML soft decision performance  相似文献   

6.
An efficient algorithm for calculating the ith bit error probability of a binary linear code over the binary symmetric channel (BSC) is presented. It is proved that the exact ith bit error probability of maximum-likelihood (ML) decoding, bounded distance decoding, and symbol-wise maximum a posteriori probability (MAP) decoding can be obtained with time complexity O(n2/sup n-k/), where n and k denote the length and the dimension of the target code. The proposed methods are applicable to any binary linear code with redundancy up to nearly 25-30 bits with a typical personal computer.  相似文献   

7.
A code C detects error e with probability 1-Q(e),ifQ(e) is a fraction of codewords y such that y, y+e/spl isin/C. We present a class of optimal nonlinear q-ary systematic (n, q/sup k/)-codes (robust codes) minimizing over all (n, q/sup k/)-codes the maximum of Q(e) for nonzero e. We also show that any linear (n, q/sup k/)-code V with n /spl les/2k can be modified into a nonlinear (n, q/sup k/)-code C/sub v/ with simple encoding and decoding procedures, such that the set E={e|Q(e)=1} of undetected errors for C/sub v/ is a (k-r)-dimensional subspace of V (|E|=q/sup k-r/ instead of q/sup k/ for V). For the remaining q/sup n/-q/sup k-r/ nonzero errors, Q(e)/spl les/q/sup -r/for q/spl ges/3 and Q(e)/spl les/ 2/sup -r+1/ for q=2.  相似文献   

8.
Burst-error-correcting algorithm for Reed-Solomon codes   总被引:4,自引:0,他引:4  
A new decoding algorithm for burst-error-correction is proposed. The proposed algorithm can effectively correct burst errors of length approaching n-k symbols for (n, k) Reed-Solomon (RS) codes. Compared with existing algorithms, the algorithm enables much faster decoding with far less computational complexity  相似文献   

9.
Any symbol in a redundant code can be recovered when it belongs to certain erasure patterns. Several alternative expressions of a given symbol, to be referred to as its replicas, can therefore be computed in terms of other ones. Decoding is interpreted as decoding upon a received symbol, given itself and a number of such replicas, expressed in terms of other received symbols. For linearq-ary (n,k)block codes, soft-decision demodulation and memoryless channels, the maximum-likelihood decision rule on a given symbol is formulated in terms ofr leqn - klinearly independent replicas from the parity-check equations. All replicas deriving from therselected replicas by linear combination are actually taken into account in this decision rule. Its implementation can be direct; use transformations or a sequential circuit implementing a trellis representation of the parity-check matrix. Ifr = n - k, decoding is optimum, in the sense of symbol-by-symbol maximum-likelihoed. Simplification results in the transformed and sequential implementations whenr < n - k. If the selected replicas are disjoint, generalized (q-ary, weighted) threshold decoding results. The decoding process can easily be modffied in order to provide word-by-word maximum-likelihood decoding. Convolutional codes are briefly considered. Two specific problems are discussed: the use of previous decisions, which leads to a weighted generalization of feedback decoding, and the extension of replication decoding to nonsystematic codes.  相似文献   

10.
There are two kinds of perfect (k-t)-deletion-correcting codes with words of length k over an alphabet of size v, those where the coordinates may be equal and those where all coordinates must be different. We call these two kinds of codes T*(t,k,v)-codes and T(t,k,v)-codes respectively. Both a T*(t,k,v)-code and a T(t,k,v)-code are capable of correcting any combination of up to (k-t) deletions and insertions of letters occurred in transmission of codewords. In this correspondence, we consider constructions for the codes from directed designs. By means of these constructions, the existence of a T(2,7,v)-code is settled for all positive integers v with the exception of 68 values of v; T*(2,7,v)-codes are constructed for all integers vges2350. A large number of explicit constructions for T*(2,7,v)-codes with v<2350 are also presented  相似文献   

11.
Optimal sectionalization of a trellis   总被引:1,自引:0,他引:1  
While the complexity of trellis decoding for a given block code is essentially a function of the number of states and branches in its trellis, the decoding complexity may be often reduced by means of an appropriate sectionalization of the trellis. Notwithstanding the many examples of such sectionalizations for particular codes that appeared in the literature, no systematic method for finding the best sectionalization of a given trellis is presently known. We present a polynomial-time algorithm which produces the optimal sectionalization of a given trellis T in time O(n2), where n is the length of the code generated by T. The algorithm is developed in a general setting of certain operations and functions defined on the set of trellises; it therefore applies to both linear and nonlinear codes, and easily accommodates a broad range of optimality criteria. The particular optimality criterion based on minimizing the total number of additions and comparisons required for maximum-likelihood trellis decoding is investigated in detail: several different methods for decoding a given trellis are discussed and compared in a number of examples. Finally, analysis of the dynamical properties of certain optimal sectionalizations is presented  相似文献   

12.
In this paper, we prove the following two results that expose some combinatorial limitations to list decoding Reed-Solomon codes. 1) Given n distinct elements alpha1,...,alphan from a field F, and n subsets S1,...,Sn of F, each of size at most l, the list decoding algorithm of Guruswami and Sudan can in polynomial time output all polynomials p of degree at most k that satisfy p(alphai)isinSi for every i, as long as ldelta for small enough delta, we exhibit an explicit received word with a superpolynomial number of Reed-Solomon codewords that agree with it on (2-epsi)k locations, for any desired epsi>0 (agreement of k is trivial to achieve). Such a bound was known earlier only for a nonexplicit center. Finding explicit bad list decoding configurations is of significant interest-for example, the best known rate versus distance tradeoff, due to Xing, is based on a bad list decoding configuration for algebraic-geometric codes, which is unfortunately not explicitly known  相似文献   

13.
In this paper, we propose a near maximum likelihood (ML) scheme for the decoding of multiple-input–multiple-output (MIMO) systems. By employing the metric-first search method, Schnorr–Euchner enumeration, and branch-length thresholds in a single frame systematically, the proposed technique provides efficiency that is higher than those of other conventional near-ML decoding schemes. From simulation results, it is confirmed that the proposed scheme has computational complexity lower than those of other near-ML decoders while maintaining the bit error rate (BER) very close to the ML performance. The proposed scheme, in addition, possesses the capability of allowing flexible tradeoffs between the computational complexity and BER performance.   相似文献   

14.
On the capacity of two-dimensional run-length constrained channels   总被引:2,自引:0,他引:2  
Two-dimensional binary patterns that satisfy one-dimensional (d, k) run-length constraints both horizontally and vertically are considered. For a given d and k, the capacity Cd,k is defined as Cd,k=limm,n→∞log2Nm,n d,k/mn, where Nm,nd,k denotes the number of m×n rectangular patterns that satisfy the two-dimensional (d,k) run-length constraint. Bounds on Cd,k are given and it is proven for every d⩾1 and every k>d that Cd,k=0 if and only if k=d+1. Encoding algorithms are also discussed  相似文献   

15.
In general, an (n,k) maximum distance separable (MDS) code over GF(pm) can not correct all burst erasures of length (n-k)m over GF(p). In this letter, we constructively show that such a linear MDS code over GF(pm) can be modified to correct all burst erasures of length up to (n-k)m over GF(p), while maintaining its MDS property over GF(pm)  相似文献   

16.
For high rate k/n convolutional codes (k/n > 0.5), a trellis based implementation of a posteriori probability (APP) decoders is less complex on the dual code trellis owing to its branch complexity (2n-k ) being lower than the code trellis (2k). The log scheme used for APP decoders is not attractive for practical implementation owing to heavy quantisation requirements. As an alternative, presented is an arc hyperbolic tangent (AHT) scheme for implementing the dual- APP decoder. The trellis based implementation of this AHT dual APP decoder is discussed and some fundamental differences between primal APP and dual APP decoders that have an effect on a quantised implementation are reported.  相似文献   

17.
Optimum soft decoding of sources compressed with variable length codes and quasi-arithmetic codes, transmitted over noisy channels, can be performed on a bit/symbol trellis. However, the number of states of the trellis is a quadratic function of the sequence length leading to a decoding complexity which is not tractable for practical applications. The decoding complexity can be significantly reduced by using an aggregated state model, while still achieving close to optimum performance in terms of bit error rate and frame error rate. However, symbol a posteriori probabilities can not be directly derived on these models and the symbol error rate (SER) may not be minimized. This paper describes a two-step decoding algorithm that achieves close to optimal decoding performance in terms of SER on aggregated state models. A performance and complexity analysis of the proposed algorithm is given.  相似文献   

18.
An error-trellis is a directed graph that represents all the sequences belonging to the coset which contains the symbol-by-symbol detected version of a given received sequence. A modular construction of error-trellises for an (n,k) convolutional code over GF(q) is presented. The trellis is designed on the basis of partitioning the scalar check matrix of the code into submatrices of l rows, accompanied with a corresponding segmentation of the syndrome. The value of the design parameter l is an essentially unconstrained multiple of n-k. For all the cosets of the code, the sections of the error-trellis are drawn from a collection of only ql modules; the module for each section is determined by the value of the associated syndrome segment. In case the construction is based on a basic polynomial check matrix, either canonical or noncanonical, then the error-trellis is minimal in the sense that σ⩽μ, where σ is the dimension of the state space of the trellis and μ is the constraint length of a canonical generator matrix for the code. For basic check matrices with delay-free columns, the inequality reduces to σ=μ  相似文献   

19.
Almost all the probabilistic decoding algorithms known for convolutional codes, perform decoding without prior knowledge of the error locations. Here, we introduce a novel maximum-likelihood decoding algorithm for a new class of convolutional codes named as the state transparent convolutional (STC) codes, which due to their properties error detection and error locating is possible prior to error correction. Hence, their decoding algorithm, termed here as the STC decoder, allows an error correcting algorithm to be applied only to the erroneous portions of the received sequence referred to here as the error spans (ESPs). We further prove that the proposed decoder, which locates the ESPs and applies the Viterbi algorithm (VA) only to these portions, always yields a decoded path in trellis identical to the one generated by the Viterbi decoder (VD). Due to the fact that the STC decoder applies the VA only to the ESPs, hence percentage of the single-stage (per codeword) trellis decoding performed by the STC decoder is considerably less than the VD, which is applied to the entire received sequence and this reduction is overwhelming for the fading channels, where the erroneous codewords are mostly clustered. Furthermore, through applying the VA only to the ESPs, the resulting algorithm can be viewed as a new formulation of the VD for the STC codes that analogous to the block decoding algorithms provides a predecoding error detection and error locating capabilities, while performing less single-stage trellis decoding.  相似文献   

20.
We construct parity-concatenated trellis codes in which a trellis code is used as the inner code and a simple parity-check code is used as the outer code. From the Tanner-Wiberg-Loeliger (1981, 1996) graph representation, several iterative decoding algorithms can be derived. However, since the graph of the parity-concatenated code contains many short cycles, the conventional min-sum and sum-product algorithms cannot achieve near-optimal decoding. After some simple modifications, we obtain near-optimal iterative decoders. The modifications include either (a) introducing a normalization operation in the min-sum and sum-product algorithms or (b) cutting the short cycles which arise in the iterative Viterbi algorithm (IVA). After modification, all three algorithms can achieve near-optimal performance, but the IVA has the least average complexity. We also show that asymptotically maximum-likelihood (ML) decoding and a posteriori probability (APP) decoding can be achieved using iterative decoders with only two iterations. Unfortunately, this asymptotic behavior is only exhibited when the bit-energy-to-noise ratio is above the cutoff rate. Simulation results show that with trellis shaping, iterative decoding can perform within 1.2 dB of the Shannon limit at a bit error rate (BER) of 4×10-5 for a block size of 20000 symbols. For a block size of 200 symbols, iterative decoding can perform within 2.1 dB of the Shannon limit  相似文献   

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