An algorithm for calculating the exact bit error probability of a binary linear code over the binary symmetric channel

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.     

Two topics on linear unequal error protection codes: Bounds on their length and cyclic code classes

It is possible for a linear block code to provide more protection for selected positions in the input message words than is guaranteed by the minimum distance of the code. Linear codes having this property are called linear unequal error protection (LUEP) codes. Bounds on the length of a LUEP code that ensures a given unequal error protection are derived. A majority decoding method for certain classes of cyclic binary UEP codes is treated. A list of short (i.e., of length less than 16) binary LUEP codes of optimal (i.e., minimal) length and a list of all cyclic binary UEP codes of length less than 40 are included.     

Bounds on the decoding error probability of binary linear codes viatheir spectra

Bounds on the error probability of maximum likelihood decoding of a binary linear code are considered. The bounds derived use the weight spectrum of the code and they are tighter than the conventional union bound in the case of large noise in the channel. The bounds derived are applied to a code with an average spectrum, and the result is compared to the random coding exponent. The author shows that the bound considered for the binary symmetrical channel case coincides asymptotically with the random coding bound. For the case of AWGN channel the author shows that Berlekamp's (1980) tangential bound can be improved, but even this improved bound does not coincide with the random coding bound, although it can be very close to it     

Bounds on the bit rate of linear runlength codes (Corresp.)

We define a class of easily implementable suboptimum codes for runlengths of binary facsimile images which we call linear runlength codes because the number of bits we transmit for each run is approximately proportional to its length. Then, assuming the runlengths are geometrically distributed, we derive bounds on the minimum bit rate achievable by linear runlength codes of a given size.     

Bounds on the worst case probability of undetected error

Upper and lower bounds on the average worst-case probability of undetected error for linear [n,k,q] codes are given     

Bounds on the probability of error for convolutional codes (Corresp.)

For the single error correcting convolutional codes introduced by Wyner and Ash, it is shown that if sufficiently few errors occur in an appropriate neighborhood of a block, the probability of correctly decoding that block is independent of errors outside that neighborhood. This fact is used to derive bounds on the bit error probability and the mean time to first error.     

On the relationship between mutual information and bit error probability for some linear dispersion codes

In this paper, we derive the relationship between the bit error probability (BEP) of maximum a posteriori (MAP) bit detection and the bit minimum mean square error (BMMSE). By using this result, the relationship between the mutual information and the BEP is derived for multiple-input multiple-output (MIMO) communication systems with the bit-linear linear-dispersion (BLLD) codes for the Gaussian channel. From the relationship, the lower and upper bounds on the mutual information can be derived.     

Bounds on the undetected error probabilities of linear codes forboth error correction and detection

The author investigates the (n, k, d⩾2t+1) binary linear codes, which are used for correcting error patterns of weight at most t and detecting other error patterns over a binary symmetric channel. In particular, for t=1, it is shown that there exists one code whose probability of undetected errors is upper-bounded by (n+1) [2^n-k-n]^-1 when used on a binary symmetric channel with transition probability less than 2/n     

Bounds on the error probability ofml decoding for block and turbo-block codes

The performance of either structured or random turbo-block codes and binary, systematic block codes operating over the additive white Gaussian noise (Awgn) channel, is assessed by upper bounds on the error probalities of maximum likelihood (Ml) decoding. These bounds on the block and bit error probability which depend respectively on the distance spectrum and the input-output weight enumeration function (Iowef) of these codes, are compared, for a variety of cases, to simulated performance of iterative decoding and also to some reported simulated lower bounds on the performance ofMl decoders. The comparisons facilitate to assess the efficiency of iterative decoding (as compared to the optimalMl decoding rule) on one hand and the tightness of the examined upper bounds on the other. We focus here on uniformly interleaved and parallel concatenated turbo-Hamming codes, and to that end theIowefs of Hamming and turbo-Hamming codes are calculated by an efficient algorithm. The usefulness of the bounds is demonstrated for uniformly interleaved turbo-Hamming codes at rates exceeding the cut-off rate, where the results are compared to the simulated performance of iteratively decoded turbo-Hamming codes with structured and statistical interleavers. We consider also the ensemble performance of ‘repeat and accumulate’ (Ka) codes, a family of serially concatenated turbo-block codes, introduced by Divsalar, Jin and McEliece. Although, the outer and inner codes possess a very simple structure: a repetitive and a differential encoder respectively, our upper bounds indicate impressive performance at rates considerably beyond the cut-off rate. This is also evidenced in literature by computer simulations of the performance of iteratively decodedRa codes with a particular structured interleaver.     

The probability of undetected error when a code is used for error correction and detection (Corresp.)

We study the probability of having an undetected error when a linear block code is used to correct up toterrors on a symmetric channel, and the remaining power of the code is used for error detection.     

Bounds on the pairwise error probability of coded DS/SSMAcommunication systems in Rayleigh fading channels

Arbitrarily tight upper and lower bounds on the pairwise error probability (PEP) of a trellis-coded or convolutional-coded direct-sequence spread-spectrum multiple-access (DS/SSMA) communication system over a Rayleigh fading channel are derived. A new set of probability density functions (PDFs) and cumulative distribution functions (CDFs) of the multiple-access interference (MAI) statistic is derived, and a modified bounding technique is proposed to obtain the bounds. The upper bounds and lower bounds together specify the accuracy of the resulting estimation of the PEP, and give an indication of the system error performance. Several suboptimum decoding schemes are proposed and their performances are compared to that of the optimum decoding scheme by the average pairwise error probability (APEP) values. The approach can be used to accurately study the multiple-access capability of the coded DS/SSMA system without numerical integrations     

Upper bound on the bit error probability of TCM overfrequency-selective Rayleigh fading channel

A new upper bound on the bit error probability using Chernoff's bound of trellis code modulation (TCM) is introduced. It can be applied to a frequency-selective Rayleigh fading channel. The bound indicates a new design criterion, which is the product of the intersymbol interference (ISI) distance     

Every binary (2m-2, 22(m)-2-m, 3) code can belengthened to form a perfect code of length 2m-1

We answer a problem posed by Etzion and Vardy (1998) by showing that a binary code of length N=2(m)-2 with 2(N-m) codewords and minimum distance three can always be lengthened to form a perfect code of length 2(m-1)     

Concurrent error detection in array multipliers forGF(2m) fields

Based on Lee-Lu-Lee's array multipliers and the RESO method, a concurrent error detection scheme in array multipliers for GF(2^m ) fields is presented and only one clock cycle is added. The fault tolerant capability in such array multipliers is also included and only two extra clock cycles are required     

Estimation of the exponent of the decoding error probability for a special generalized LDPC code

A special construction of a generalized low-density parity-check (LDPC) code and a low-complexity algorithm for his code decoding are proposed. A lower estimate of the exponent of the decoding error probability is obtained for the considered code and the decoding algorithm. This estimate leads the conclusion that, in an ensemble of considered LDPC codes, there are codes with rates as high as the code capacity and the exponent of the decoding error probability exceeds zero.     

On some properties of the undetected error probability of linear codes (Corresp.)

A recent paper [1] discussed the2^{-p}bound (wherep = n- k) for the probability of undetected errorP(epsilon)for an(n,k)block code used for error detection on a binary symmetric channel. This investigation is continued and extended and dual codes are studied. The dual and extension of a perfect code obey the2^{-p}bound, but this is not necessarily true for arbitrary codes that obey the bound. Double-error-correcting BCH codes are shown to obey the bound. Finally the problem of constructing uniformly good codes is examined.     

Improved bounds on the word error probability of RA(2) codes with linear-programming-based decoding

This paper deals with the linear-programming-based decoding algorithm of Feldman and Karger for repeat-accumulate "turbo-like" codes. We present a new structural characterization that captures the event that decoding fails. Based on this structural characterization, we develop polynomial algorithms that, given an RA(2) code, compute upper and lower bounds on the word error probability P/sub w/ for the binary-symmetric and the additive white Gaussian noise (AWGN) channels. Our experiments with an implementation of these algorithms for bounding P/sub w/ demonstrate in many interesting cases an improvement in the upper bound on the word error probability by a factor of over 1000 compared to the bounds by Feldman et al.. The experiments also indicate that the improvement in upper bound increases as the codeword length increases and the channel noise decreases. The computed lower bounds on the word error probability in our experiments are roughly ten times smaller than the upper bound.     

Bounds on the error probability of FSK and PSK receivers due to non-Gaussian noise in fading channels

This paper derives upper and lower bounds on binary error probability as a function of signal-to-noise ratio for several digital data systems operating over a complex Gaussian fading channel. Bounds of varying degrees of tightness are obtained by placing certain physically meaningful constraints on the allowable detected noise probability distributions. Detailed derivations are given for bounds corresponding to constraints on the "crest-factor" of the detected noise and on the ratio of peak noise power to average signal power. The calculations include the effects of diversity corn-billing and are applicable to frequency-shift keying (FSK), binary and quaternary phase-shift keying (PSK) using a pilot tone as reference, and binary and quaternary PSK using the previous signaling element as a phase reference. It is an interesting result of this paper that for moderate noise crest factors the upper and lower bounds can be rather close. In the particular case of nondiversity operation, for example, the lower bound actually becomes approximately equal to the upper bound at SNR's of30dB or greater for noises with a crest factor as high as20dB.     

A bit error probability analysis of a digital PLL based demodulatorof differentially encoded BPSK and QPSK modulation

Presents a bit error probability analysis of a digital phase-locked loop based demodulator, of differentially encoded BPSK and QPSK modulations. Differential decoding is a method of resolving a phase ambiguity, typical of fully modulated signals, that uses two consecutive demodulated symbols to estimate the information symbols. The effects of a noisy phase reference on demodulator performance are well documented for uncoded modulations (single symbol demodulation). The paper investigates performance for phase reference time variations between the two symbols. The time varying reference investigated is produced by a digital phase-locked loop. The noisy phase reference has negligible additional effect on the bit error probability for differentially encoded BPSK and QPSK     

Optimum(n,2)codes for small values of channel error probability (Corresp.)

The generator matrices of(n, 2)codes which are optimum for the BSC with sufficiently small channel error probabilitypare given.