Probability Of Undetected Error Research Articles

On computing undetected error probabilities on the Gilbert channel

Two methods for computing the probability of undetected error on the Gilbert (1960) channel are examined. First, using a method proposed by Kittel (1978), we study some standard cyclic redundancy codes and compare the results with those on the binary symmetric channel. Then we consider a general method of approximate code evaluation, proposed by Elliott, which involves P(m, n), the probability of m errors in a block of length n bits. A nonrecursive technique for computing P(m, n) on the Gilbert channel is described.

On the undetected error probability of nonlinear binary constant weight codes

The undetected error probability (UEP) of binary (n, 2/spl delta/, m) nonlinear constant weight codes over the binary symmetric channel (BSC) is investigated, where n is the blocklength, m is the weight of codeword and 2/spl delta/ is the minimum distance of the codes. The distance distribution of the (n, 2, m) nonlinear constant weight codes is evaluated. It is proven in this paper that the (5, 2, 2) code, (5, 2, 3) code, (6, 2, 3) code, (7, 2, 4) code, (7, 2, 3) code and (8, 2, 4) code are the only proper error-detecting codes in the (n, 2, m) nonlinear constant weight codes for n/spl ges/5, in the sense that their UEP is increased monotonically with the channel error rate p, of course all these proper codes are m-out-of-n codes. Furthermore, it is conjectured that except for the cases of n/spl les/4/spl delta/, there are no proper error-detecting binary (n, 2/spl delta/, m) nonlinear constant weight codes, for n>8 and /spl deltaspl ges/1.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A robust rate-adaptive hybrid ARQ scheme for frequency-hopped spread-spectrum multiple-access communication systems

This paper considers the application of rate-adaptive coding (RAC) to a spread spectrum multiple access (SSMA) communication system. Specifically, RAC using a variable rate Reed-Solomon (RS) code with a single decoder is applied to frequency-hopped SSMA. We show that this combination can accommodate a larger number of users compared to that with conventional fixed-rate coding. This increase is a result of a reduction in the channel interference from other users. The penalty for this improvement in most cases is a slight increase in the delay (composed of propagation and decoding delay). The throughput and the undetected error probability for a M-ary symmetric channel are analyzed, and performance results are presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Error-detecting codes and undetected error probabilities

The definition of good codes for error-detection is given. It is proved that a (n, k) linear block code in GF(q) are the good code for error-detection, if and only if its dual code is also. A series of new results about the good codes for error-detection are derived. New lower bounds for undetected error probabilities are obtained, which are relative ton andk only, and not the weight structure of the codes.

Special hardware for computing the probability of undetected error for certain binary CRC codes and test results

A hardware device for efficiently evaluating the probability of undetected error for a class of CRC error detection codes with a large number of parity check digits is described. The generator polynomial for the codes in this class are of the form g(x)=(1+x)p(x) where p(x) is a primitive irreducible polynomial. The degree of g(x), R, is the number of parity check digits. Using this hardware, a search was conducted for codes in this class (for 8/spl les/R/spl les/39) which are proper for shortened block lengths. A table of codes satisfying this condition is included. >

Frequency-hopped ARQ for wireless network data services

The throughput efficiency and probability of undetected error of automatic-repeat-request (ARQ) protocols employing slow frequency hopping over Gilbert-Elliott (1960, 1963) fading channels are investigated. Throughput efficiency formulas for the three basic ARQ schemes, stop-and-wait (SW), go-back-N (GBN), and selective-repeat (SR), are obtained, and the improvement in throughput efficiency due to slow frequency hopping is examined. Results are also presented on the problem of determining the number of carriers used for frequency hopping so as to minimize the effect of fading channel memory and, therefore, optimize system performance. This work has application to TDMA architectures and packet data services for third generation wireless networks, where projected data service requirements call for error rates on the order of 10/sup -9/, and the multiple service, multiple access environment demands that the delay for any service be minimized. >

The single burst error detection performance of binary cyclic codes

Using a new statistical model for burst errors, the authors calculate the conditional probability of undetected error for CRC codes with generator polynomials of the form g(x)=(1+x)p(x), p(x) a primitive polynomial. They show that the choice of p(x) affects the performance of these codes. >

Performance evaluation of the generalized type-II hybrid ARQ scheme with noisy feedback on Markov channels

The generalized type-II hybrid automatic repeat request (ARQ) scheme on Markov channels is examined in order to study the effect of feedback channel errors on the performance of ARQ systems. It is shown that it is possible to derive expressions for certain critical performance parameters, such as the throughput efficiency, the probabilities of packet loss, undetected error, and correct delivery. To provide a means of comparison, a parallel set of expressions is provided under the usual assumption of an error-free feedback channel. By use of simulations, the ARQ system performance is examined under noiseless feedback and noisy feedback. It is found that feedback channel noise can result in the loss of packets, an increase in the number of undetected errors, and the occurrence of unnecessary transmissions. To enhance the performance of the GH-II ARQ scheme, a predictor is used and found to lower the probability of undetected error, reduce the number of unnecessary transmissions, and increase the throughput efficiency. >

Hybrid ARQ schemes for point-to-multipoint communication over nonstationary broadcast channels

Hybrid automatic-repeat-request (ARQ) error control schemes make use of both error detection and error correction in order to achieve high throughput and low undetected error probabilities on two way channels. Two hybrid ARQ schemes, termed hybrid go-back-N (HGB-N) and hybrid selective-repeat (HSR), are proposed for point-to-multipoint communications over broadcast channels. Both schemes incorporate a concatenated code for error correction and error detection. The performance study of the hybrid schemes is based on a two-state Markov model of a burst noise channel. An analytic solution is derived for the throughput efficiency of the HSR scheme, while approximations and computer simulation are used to evaluate the throughput efficiency of the HGB-N scheme. It is shown that the schemes perform considerably better than the corresponding pure ARQ schemes in which a block code is used for error detection only, especially in environments with a large number of receivers and large channel roundtrip delays, such as satellite broadcast links.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Computing the probability of undetected error for shortened cyclic codes

The authors present a general technique for computing P/sub e/ for all possible shortened versions of cyclic codes generated by any given polynomial. The technique is recursive, i.e. computes P/sub e/ for a given code block length n from that of the code block length n-1. The proposed computation technique for determining P/sub e/ does not require knowledge of the code weight distributions. For a generator polynomial of degree r, and mod g mod nonzero coefficients, the technique yields P/sub e/ for all code block lengths up to length n in time complexity O(n mod g mod 2/sup r+ mod g mod /). Channels with variable bit error probabilities can be analyzed with the same complexity. This enables the performance of the code generator polynomials to be analyzed for burst errors. >

Optimal codes for error detection

The probability of undetected error for codes over GF(2/sup m/) having a generator matrix over GF(2) is studied. The optimal codes of dimensions four or less are determined. For higher dimensions, some properties of optimal codes are determined.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Decoding strategies for Reed-Solomon product codes: application to digital video recording systems

The authors propose two decoding strategies for Reed-Solomon product codes, and analyze their performance on a random error channel. The performance measures of interest are the probability of undetected error at the inner decoder output and that of decoding failure at the outer decoder output. It has been shown that the proposed decoding strategies give a significant performance improvement over the conventional decoder.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On the undetected error probability of triple-error-correcting BCH codes

The probability of undetected error P/sub u/( epsilon ) for the primitive triple-error-correcting BCH codes of blocklength 2/sup m/-1 used solely for error detection on a binary symmetric channel with crossover probability epsilon <or=1/2 is examined. It is shown that for odd values of m, P/sub u/( epsilon ) increases monotonically with epsilon . For even values of m, this is not necessarily true. However, for a fixed epsilon , as m increases, P/sub u/( epsilon ) approaches 2/sup -p/, where p is the number of parity bits. Similar characteristics are exhibited by the extended codes.<<ETX>>

On the monotonic property of the probability of undetected error for a shortened code

The monotonic property of the probability of undetected error is considered when a shortened code of a linear code over GF(q) is used for error detection on a q-ary symmetric channel. Some conditions are presented under which the probability of undetected error is (or is not) monotonic with respect to the code length or the symbol error probability. It is shown that the probability of undetected error for a maximum distance separable code is monotonic with respect to the codelength. It is also shown that the probability of undetected error of a shortened code of any binary cyclic Hamming code generated by a trinomial is not monotonic with respect to the bit-error rate if the code length is short.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The performance analysis of a concatenated ARQ scheme using parity retransmissions

The authors analyze an ARQ (automatic repeat request) scheme using parity retransmissions. By dividing each transmission block into subblocks, the combination of a transmission block and its retransmission block has the structure of a concatenated code. They derive formulae for calculating an upper bound on the probability of undetected errors and a lower bound on the throughput for such an ARQ scheme. Numerical results show that even with extremely simple code design, such an ARQ scheme has satisfactory performance.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Undetected error probabilities of codes for single-error correction and error detection

It is shown that the probabilities of undetected errors for maximum-length codes are upper bounded by (n + 1). 2−(n − k), if the codes are used for correcting every single error and detecting other errors over a binary symmetric channel of which the transition probability is less than 1/2. Moreover, it is shown that binary linear codes with poor distance property do not satisfy the aforementioned bound.

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

The author investigates the (n, k, d>or=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/sup n-k/-n)/sup -1/ when used on a binary symmetric channel with transition probability less than 2/n.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Optimum cycle redundancy-check codes with 16-bit redundancy

Optimal binary cyclic redundancy-check codes with 16 parity bits (CRC-16 codes) are presented and compared to those in existing standards for minimum-distance, undetected-error probability on binary symmetric channels (BSCs) and properness. The codes in several cases are seen to be superior at block lengths of practical interest when they are used on low-noise BSCs. The optimum minimum distance obtainable by some CRC-16 codes is determined for all block lengths. For several typical low-noise BSCs the minimum undetected error probability achievable with some CRC-16 codes is given for all block lengths.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Error characteristics of fiber distributed data interface (FDDI)

An analysis is made of the impact of various design decisions on the error detection capability of the fiber distributed data interface (FDDI), a 100-Mb/s fiber-optic LAN standard being developed by the American National Standards Institute (ANSI). In particular, the frame error rate, token loss rate, and undetected error rate are quantified. Several characteristics of the 32-b frame check sequence (FCS) polynomial, which is also used in IEEE 802 LAN protocols, are discussed. The standard uses a nonreturn to zero invert on ones (NRZI) signal encoding and a 4-b to 5-b (4b/5b) symbol encoding in the physical layer. Due to the combination of NRZI and 4b/5b encoding, many noise events are detected by code (or symbol) violations. A large percentage of errors are detected by FCS violations. The errors that escape these three violations remain undetected. The probability of undetected errors due to creation of false starting delimiters, false ending delimiters, or merging of two frames is analyzed. It is shown that every noise event results in two code bit errors, which in turn may result in up to four data bit errors. The FCS can detect up to two noise events. Creation of a false starting delimiter or ending delimiter on a symbol boundary also requires two noise events. This assumes enhanced frame validity criteria. The author justifies the enhancements by quantifying their effect.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Parity retransmission hybrid ARQ using rate 1/2 convolutional codes on a nonstationary channel

A parity retransmission hybrid automatic repeat request (ARQ) scheme is proposed which uses rate 1/2 convolutional codes and Viterbi decoding. A protocol is described which is capable of achieving higher throughputs than previously proposed parity retransmission schemes. The performance analysis is based on a two-state Markov model of a nonstationary channel. This model constitutes a first approximation to a nonstationary channel. The two-state channel model is used to analyze the throughput and undetected error probability of the protocol presented when the receiver has both an infinite and a finite buffer size. It is shown that the throughput improves as the channel becomes more bursty.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>