Decoder malfunction in BCH decoders

A t-error-correcting bounded-distance decoder either produces the codeword nearest the received vector (if there is a codeword at distance no more than t) or indicates that no such codeword exists. However, BCH decoders based on the Peterson-Gorenstein-Zierler algorithm or the Euclidean algorithm can malfunction and produce output vectors that are not codewords at all. For any integer i no greater than t/2, if the received vector is at distance at most t-2i from a codeword belonging to a (t-i)-error-correcting BCH supercode, then the BCH decoder output is that codeword from the supercode     

The (d,k) subcode of a linear block code

A simple technique employing linear block codes to construct (d,k) error-correcting block codes is considered. This scheme allows asymptotically reliable transmission at rate R over a BSC channel with capacity C_BSC provided R ⩽C_d,k-(1+C_BSC), where C_d,k is the maximum entropy of a (d,k ) source. For the same error-correcting capability, the loss in code rate incurred by a multiple-error correcting (d,k) code resulting from this scheme is no greater than that incurred by the parent linear block code. The single-error correcting code is asymptotically optimal. A modification allows the correction of single bit-shaft errors as well. Decoding can be accomplished using off-the-shelf decoders. A systematic (but suboptimal) encoding scheme and detailed case studies are provided     

An improvement to generalized-minimum-distance decoding

A novel acceptance criterion that is less stringent than previous criteria is developed. The criterion accepts the codeword that is closest to the received vector for many cases where previous criteria fail to accept any codeword. As a result, the performance of generalized minimum distance (GMD) decoding is better if the new criterion is used. For M-ary signaling, the weights used in GMD decoding are generalized to permit each of the possible M symbol values to have a different weight     

A burst-error-correcting algorithm for Reed-Solomon codes

It is known that for a burst-error environment, the error-correcting capability of Reed-Solomon codes can be extended beyond the Singleton bound with a high degree of confidence. This is significant in that an (n, k) code with an arbitrarily small probability of a miscorrection can correct more than (n- k)/2 errors. A decoding algorithm for correcting a burst of length greater than (n-k)/2 is presented     

Decoding algorithms with input quantization and maximum errorcorrection capability

Decoding that uses soft-decision information but with multiple low-complexity decoders are investigated. These decoders correct only errors and erasures. The structure of the receiver consists of a bank of z demodulators followed by errors- and erasures-correcting decoders operating in parallel. Each demodulator has a threshold for determining when to erase a given symbol. We assign a cost f(&thetas;) to the noise for causing an erasure when the receiver uses a particular threshold &thetas; and a (larger) cost f(&thetas;¯) for causing an error. The goal in designing the receiver is to choose the thresholds to maximize the noise cost which is necessary to cause a decoding error. We demonstrate that the above formulation is solvable for many channels including the M-ary input-output channel, the additive channel with coherent demodulation, and an additive channel with orthogonal modulation and noncoherent demodulation. Then we show that the maximum worst case error-correcting capability of the parallel decoding algorithms is the same as the maximum worst case error-correcting capability of a correlation decoder with the same number of quantization regions     

A reduced-complexity sequence detector with soft outputs forpartial-response channels

Data transmission in a binary partial-response channel is often accomplished using a concatenated code consisting of an inner modulation code and an outer error-correcting code (ECC). We consider two inner decoders for such a code, each consisting of a reduced-complexity sequence detector modified to provide an estimate of the reliability of each bit. These reliability values are then used by the outer decoder to achieve improved performance. Although one of these decoders is considerably simpler than the other, their performances were comparable in the cases we considered. Both achieve considerable improvement over a decoder that uses hard-decision decoding of the inner code     

A convolutional single-parity-check concatenated coding scheme forhigh-data-rate applications

The coding scheme uses a set of n convolutional codes multiplexed into an inner code and a (n,n-1) single-parity-check code serving as the outer code. Each of the inner convolutional codes is decoded independently, with maximum-likelihood decoding being achieved using n parallel implementations of the Viterbi algorithm. The Viterbi decoding is followed by additional outer soft-decision single-parity-check decoding. Considering n=12 and the set of short constraint length K=3, rate 1/2 convolutional codes, it is shown that the performance of the concatenated scheme is comparable to the performance of the constraint length K=7, rate 1/2 convolutional code with standard soft-decision Viterbi decoding. Simulation results are presented for the K=3, rate 1/2 as well as for the punctured K=3, rate 2/3 and rate 3/4 inner convolutional codes. The performance of the proposed concatenated scheme using a set of K=7, rate 1/2 inner convolutional codes is given     

Neural networks, error-correcting codes, and polynomials over thebinary n-cube

Several ways of relating the concept of error-correcting codes to the concept of neural networks are presented. Performing maximum-likelihood decoding in a linear block error-correcting code is shown to be equivalent to finding a global maximum of the energy function of a certain neural network. Given a linear block code, a neural network can be constructed in such a way that every codeword corresponds to a local maximum. The connection between maximization of polynomials over the n-cube and error-correcting codes is also investigated; the results suggest that decoding techniques can be a useful tool for solving such maximization problems. The results are generalized to both nonbinary and nonlinear codes     

Short codes with a given covering radius

The covering radius r of a code is the maximum distance from any vector in the space containing the code to the nearest codeword. The authors introduce a new function l(m,r), called the length function, which equals the smallest length of a binary code of codimension m and covering radius r. They investigate basic properties of the length function. Projective geometries over larger fields are used to construct families of codes which improve significantly the upper bound for l(m,2) obtained by amalgamation of Hamming codes. General methods are developed for ruling out the existence of codes of covering radius 2 with a given codimension and length resulting in lower bounds for l(m,2). A table is presented which gives the best results now known for l(m,r) with m⩽12 and r⩽12     

Sequential decoding based on an error criterion

An analysis of sequential decoding is presented that is based on the requirement that a set probability error P_e be achieved. The error criterion implies a bounded tree or trellis search region: the shape of this is calculated for the case of a binary symmetric channel with crossover probability P and random tree codes of rate R. Since the search region is finite at all combinations of p and R below capacity, there is no cutoff rate phenomenon for any P_e>0. The decoder delay (search depth), the path storage size, and the number of algorithm steps for several tree search methods are calculated. These include searches without backtracking and backtracking searches that are depth- and metric-first. The search depth of the non-backtracking decoders satisfies the Gallager reliability exponent for block codes. In average paths searched, the backtracking decoders are much more efficient, but all types require the same peak storage allocation. Comparisons are made to well-known algorithms     

An efficient algorithm for testing immutability of variable-lengthcodes

Immutable codes, which have recently been introduced as a tool for preventing undesirable changes of data recorded over write-once memories, are considered. The have the property that any change of recorded information over such memories can be detected. A fast algorithm for testing whether a variable-length code is immutable is presented. The complexity of the algorithm is O(L²), where L is the sum of the codeword lengths     

Bounds and constructions for codes correcting unidirectional errors

A brief introduction is given on the theory of codes correcting unidirectional errors, in the context of symmetric and asymmetric error-correcting codes. Upper bounds on the size of a code of length n correcting t or fewer unidirectional errors are then derived. Methods in which codes correcting up to t unidirectional errors are constructed by expurgating t-fold asymmetric error-correcting codes or by expurgating and puncturing t -fold symmetric error-correcting codes are also presented. Finally, tables summarizing some results on the size of optimal unidirectional error-correcting codes which follow from these bounds and constructions are given     

Packet error probabilities in frequency-hopped spread-spectrumpacket radio networks-memoryless frequency-hopping patterns considered

The packet error probability induced in a frequency-hopped spread-spectrum packet radio network is computed. The frequency spectrum is divided into q frequency bins. Each packet is exactly one codeword from an (M, L) Reed-Solomon code [M=number of codeword symbols (bytes); L=number of information symbols (bytes)]. Every user in the network sends each of the M bytes of his packet at a frequency chosen among the q frequencies with equal probability and independently of the frequencies chosen for other bytes (i.e., memoryless frequency-hopping patterns). Statistically independent frequency-hopping patterns correspond to different users in the network. Provided that K users have simultaneously transmitted their packets on the channel and a receiver has locked on to one of these K packets, the probability that this packet is not decoded correctly is evaluated. It is also shown that although memoryless frequency-hopping patterns are utilized, the byte errors at the receiver are not statistically independent; instead they exhibit a Markovian structure     

Repeated convolutional codes for high-error-rate channels

An error-correction scheme for an M-ary symmetric channel (MSC) characterized by a large error probability p_e is considered. The value of p_e can be near, but smaller than, 1-1/M, for which the channel capacity is zero, such as may occur in a jamming environment. The coding scheme consists of an outer convolutional code and an inner repetition code of length m that is used for each convolutional code symbol. At the receiving end, the m inner code symbols are used to form a soft-decision metric, which is passed to a soft-decision decoder for the convolutional code. The effect of finite quantization and methods to generate binary metrics for M>2 are investigated. Monte Carlo simulation results are presented. For the binary symmetric channel (BSC), it is shown that the overall code rate is larger than 0.6R₀, where R₀ is the cutoff rate of the channel. New union bounds on the bit error probability for systems with a binary convolutional code on 4-ary and 8-ary orthogonal channels are presented. For a BSC and a large m, a method is presented for BER approximation based on the central limit theorem     

New ternary line codes based on trellis structure

The trellis coding technique is applied to line-coded baseband digital transmission systems. For R=n/n+1(n=1,2,3) coding rates, a new codeword assignment model is proposed to accomplish basic requirements for line coding in which each length n binary data sequence is encoded into a length n+1 ternary (+,0,-) line codeword chosen among the code alphabet with 2ⁿ⁺² elements. Assuming Viterbi decoding, the system error performance is improved by increasing the free Euclidean distance between coded sequences. A new algorithm is given for the calculation of the free distance between line-coded sequences so obtained. For R=1/2 and R=3/4 rates, the analytical error performance upper bounds are derived. The power spectral densities of the new line codes are also calculated and compared with those of known line codes     

The capacity of binary channels that use linear codes and decoders

This paper analyzes the performance of concatenated coding systems operating over the binary-symmetric channel (BSC) by examining the loss of capacity resulting from each of the processing steps. The techniques described in this paper allow the separate evaluation of codes and decoders and thus the identification of where loss of capacity occurs. They are, moreover, very useful for the overall design of a communications system, e.g., for evaluating the benefits of inner decoders that produce side information. The first two sections of this paper provide a general technique (based on the coset weight distribution of a binary linear code) for calculating the composite capacity of the code and a BSC in isolation. The later sections examine the composite capacities of binary linear codes, the BSC, and various decoders. The composite capacities of the (8,4) extended Hamming, (24, 12) extended Golay, and (48, 24) quadratic residue codes appear as examples throughout the paper. The calculations in these examples show that, in a concatenated coding system, having an inner decoder provide more information than the maximum-likelihood (ML) estimate to an outer decoder is not a computationally efficient technique, unless generalized minimum-distance decoding of an outer code is extremely easy. Specifically, for the (8,4) extended Hamming and (24, 12) extended Golay inner codes, the gains from using any inner decoder providing side information, instead of a strictly ML inner decoder, are shown to be no greater than 0.77 and 0.34 dB, respectively, for a BSC crossover probability of 0.1 or less, However, if computationally efficient generalized minimum distance decoders for powerful outer codes, e.g., Reed-Solomon codes, become available, they will allow the use of simple inner codes, since both simple and complex inner codes have very similar capacity losses     

A concatenated coded modulation scheme for error control

A concatenated coded modulation scheme is presented for error control in data communications. The scheme is achieved by concatenating a Reed-Solomon outer code and a bandwidth efficient block inner code for M-ary phase-shift keying (PSK) modulation. Error performance of the scheme is analyzed for an additive white Gaussian noise (AWGN) channel. It is shown that extremely high reliability can be attained by using a simple M-ary PSK modulation inner-code and a relatively powerful Reed-Solomon outer code. Furthermore, if an inner code of high effective rate is used, the bandwidth expansion required by the scheme due to coding will be greatly reduced. The scheme is particularly effective for high-speed satellite communications for large file transfer where high reliability is required. A simple method is also presented for constructing block codes for M-ary PSK modulation. Soome short M-ary PSK codes with good minimum squared Euclidean distance are constructed. These codes have trellis structure and hence can be decoded with a soft-decision Viterbi decoding algorithm. Furthermore, some of these codes are phase invariant under multiples of 45° rotation     

Self-synchronizing codes derived from binary cyclic codes

The sensitivity of a binary block code to loss of synchronism (misplacement of the "commas" separating codewords) can be characterized by a pair of numbers[s, delta]such that any synchronization slip of s bits or less produces an overlap sequence differing from a legitimate codeword in at leastdeltaplaces. This definition is broader than that of comma freedom of indexdelta, which is included as the special case of s equal to the integer part of half the code block length. For codes having the slip-detecting characteristic[s, delta]there exists the possibility of implementation to restore synchronism during an interval relatively free from bit errors. It is shown that certain error-correcting binary cyclic block codes can be altered to obtain the characteristic[s, delta]by the addition of a fixed binary vector to each codeword prior to transmission. These altered cyclic codes retain the full error-correcting power of the original cyclic codes. An error-detecting/correcting data format providing protection against the acceptance of misframed data is thus obtained without the insertion of special synchronizing sequences into the bit stream.     

DC-free error-correcting codes based on convolutional codes

A new construction of direct current (DC)-free error-correcting codes based on convolutional codes is proposed. The new code is constructed by selecting a proper subcode from a convolutional code composed of two different component codes. The encoder employs a Viterbi algorithm as the codeword selector so that the selected code sequences satisfy the DC constraint. A lower bound on the free distance of such codes is proposed, and a procedure for obtaining this bound is presented. A sufficient condition for these codes to have a bounded running digital sum (RDS) is proposed. Under the assumption of a simplified codeword selection algorithm, we present an upper bound on the maximum absolute value of the RDS and derive the sum variance for a given code. A new construction of standard DC-free codes, i.e., DC-free codes without error-correcting capability, is also proposed. These codes have the property that the decoder can be implemented by simple symbol-by-symbol hard decisions. Finally, under the new construction, we propose several codes that are suitable for the systems that require small sum variance and good error-correction capability     

A general decoding technique applicable to replicated filedisagreement location and concatenated code decoding

Code symbols are treated as vectors in an r-dimensional vector space F^r over a field F. Given any ( n, k) linear block code over F with minimum distance d, it is possible to derive an (n, k) code with symbols over F^r, also with minimum distance d, which can correct any pattern of d-2 or fewer symbol errors for which the symbol errors as vectors are linearly independent. This is about twice the bound on the number of errors guaranteed to be correctable. Furthermore, if the error vectors are linearly dependent and d-2 or fewer in number, the existence of dependence can always be detected. A decoding techinque is described for which complexity increases no greater than as n ³, for any choice of code. For the two applications considered, situations are described where the probability of the error patterns being linearly dependent decreases exponentially with r