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Showing papers in "IEEE Transactions on Communications in 1984"


Journal ArticleDOI
TL;DR: It is shown that the FM capture phenomenon with slotted ALOHA greatly improves the expected progress over the system without capture due to the more limited area of possibly interfering terminals around the receiver.
Abstract: In multihop packet radio networks with randomly distributed terminals, the optimal transmission radii to maximize the expected progress of packets in desired directions are determined with a variety of transmission protocols and network configurations. It is shown that the FM capture phenomenon with slotted ALOHA greatly improves the expected progress over the system without capture due to the more limited area of possibly interfering terminals around the receiver. The (mini)slotted nonpersistent carrier-sense-multiple-access (CSMA) only slightly outperforms ALOHA, unlike the single-hop case (where a large improvement is available), because of a large area of "hidden" terminals and the long vulnerable period generated by them. As an example of an inhomogeneous terminal distribution, the effect of a gap in an otherwise randomly distributed terminal population on the expected progress of packets crossing the gap is considered. In this case, the disadvantage of using a large transmission radius is demonstrated.

1,367 citations


Journal ArticleDOI
TL;DR: This paper describes how the conflict can be resolved with partial string matching, and reports experimental results which show that mixed-case English text can be coded in as little as 2.2 bits/ character with no prior knowledge of the source.
Abstract: The recently developed technique of arithmetic coding, in conjunction with a Markov model of the source, is a powerful method of data compression in situations where a linear treatment is inappropriate. Adaptive coding allows the model to be constructed dynamically by both encoder and decoder during the course of the transmission, and has been shown to incur a smaller coding overhead than explicit transmission of the model's statistics. But there is a basic conflict between the desire to use high-order Markov models and the need to have them formed quickly as the initial part of the message is sent. This paper describes how the conflict can be resolved with partial string matching, and reports experimental results which show that mixed-case English text can be coded in as little as 2.2 bits/ character with no prior knowledge of the source.

1,318 citations


Journal ArticleDOI
TL;DR: The cost of a number of sequential coding search algorithms is analyzed in a systematic manner and it is found that algorithms that utilize sorting are much more expensive to use than those that do not; metric-first searching regimes are less efficient than breadth-first or depth-first regimes.
Abstract: The cost of a number of sequential coding search algorithms is analyzed in a systematic manner. These algorithms search code trees, and find use in data compression, error correction, and maximum likelihood sequence estimation. The cost function is made up of the size of and number of accesses to storage. It is found that algorithms that utilize sorting are much more expensive to use than those that do not; metric-first searching regimes are less efficient than breadth-first or depth-first regimes. Cost functions are evaluated using experimental data obtained from data compression and error correction studies.

623 citations


Journal ArticleDOI
TL;DR: The theory is formulated in a general manner which allows for significant freedom in the receiver modeling and the statistics of the acquisition time for the single-dwell, N-Dwell, and single- dwell systems are shown to be special cases of this unified approach.
Abstract: The purpose of this two-part paper is threefold: 1) Part I discusses the code-acquistion problem in some depth and 2) also provides a general extension to the approach of analyzing serial-search acquisition techniques via transform-domain flow graphs; 3) Part II illustrates the applicability of the proposed theoretical framework by evaluating a matchedfilter (fast-decision rate) noncoherent acquisition receiver as an example. The theory is formulated in a general manner which allows for significant freedom in the receiver modeling. The statistics of the acquisition time for the single-dwell [2], [3] and N -dwell [5] systems are shown to be special cases of this unified approach.

575 citations


Journal Article
TL;DR: A general approach is presented for designing efficient blind equalizers for one and two independent carrier transmission systems; a special algorithm is given for the CCITT V29 constellation.
Abstract: Blind equalizers do not require any known training sequence for the startup period, but can rather perform at any time the equalization directly on the data stream. In this paper, a general approach is presented for designing efficient blind equalizers for one and two independent carrier transmission systems; a special algorithm is given for the CCITT V29 constellation.

471 citations


Journal ArticleDOI
TL;DR: An efficient single-pass adaptive bandwidth compression technique using the discrete cosine transform is described, demonstrating excellent results for coding of color images at 0.4 bits/pixel corresponding to real-time color television transmission over a 1.5 Mbit/s channel.
Abstract: An efficient single-pass adaptive bandwidth compression technique using the discrete cosine transform is described. The coding process involves a simple thresholding and normalization operation on the transform coefficients. Adaptivity is achieved by using a rate buffer for channel rate equalization. The buffer status and input rate are monitored to generate a feedback normalization factor. Excellent results are demonstrated for coding of color images at 0.4 bits/pixel corresponding to real-time color television transmission over a 1.5 Mbit/s channel.

405 citations


Journal ArticleDOI
TL;DR: The high-rate punctured codes of rates 2/3 through 13/14 are derived from rate 1/2 specific convolutional codes with maximal free distance based on their bit error rate performances under soft decision Viterbi decoding.
Abstract: The high-rate punctured codes of rates 2/3 through 13/14 are derived from rate 1/2 specific convolutional codes with maximal free distance. Coding gains of derived codes are compared based on their bit error rate performances under soft decision Viterbi decoding.

399 citations


Journal ArticleDOI
M. Lema1, O. Mitchell1
TL;DR: In this paper, a color image coding system that uses absolute moment block truncation coding of luminance and chroma information is presented, which shows reasonable performance with bit rates as low as 2.13 bits/pixel.
Abstract: A new quantization method that uses the criterion of preserving sample absolute moments is presented. This is based on the same basic idea for block truncation coding of Delp and Mitchell but it is simpler in any practical implementation. Moreover, output equations are those for a two-level nonparametric minimum mean square error quantizer when the threshold is fixed to the sample mean. The application of this method to single frame color images is developed. A color image coding system that uses absolute moment block truncation coding of luminance and chroma information is presented. Resulting color images show reasonable performance with bit rates as low as 2.13 bits/pixel.

356 citations


Journal ArticleDOI
TL;DR: The results illustrate the dynamic dependence of the mean acquisition time on system parameters, such as the predetection signal-to-noise ratio (SNR), the decision threshold settings, and the ratio of the decision rate to the code rate.
Abstract: The unified theory developed in Part I [1] is employed here in the analysis of a noncoherent, matched-filter (fast-decision-rate) code acquisition receiver in a direct-sequence spread-spectrum system. The results illustrate the dynamic dependence of the mean acquisition time on system parameters, such as the predetection signal-to-noise ratio (SNR), the decision threshold settings, and the ratio of the decision rate to the code rate.

321 citations


Journal ArticleDOI
R. Nelson1, Leonard Kleinrock
TL;DR: The throughput of the network is a strictly increasing function of the receiver's ability to capture signals, and depends on the transmission range of the terminals and their probability of transmitting packets.
Abstract: In this paper we determine throughput equations for a packet radio network where terminals are randomly distributed on the plane, are able to capture transmitted signals, and use slotted ALOHA to access the channel. We find that the throughput of the network is a strictly increasing function of the receiver's ability to capture signals, and depends on the transmission range of the terminals and their probability of transmitting packets. Under ideal circumstances, we show the expected fraction of terminals in the network that are engaged in successful traffic in any slot does not exceed 21 percent.

230 citations


Journal ArticleDOI
TL;DR: A nonlinear temporal filtering algorithm using motion compensation for reducing noise in image sequences is shown to be successful in improving image quality and also improving the efficiency of subsequent image coding operations.
Abstract: Noise in television signals degrades both the image quality and the performance of image coding algorithms. This paper describes a nonlinear temporal filtering algorithm using motion compensation for reducing noise in image sequences. A specific implementation for NTSC composite television signals is described, and simulation results on several video sequences are presented. This approach is shown to be successful in improving image quality and also improving the efficiency of subsequent image coding operations.

Journal ArticleDOI
F. Natali1
TL;DR: This paper presents a brief discussion of eight different AFC loop implementations, including the block diagram, discriminator characteristic, and loop tracking error in the presence of gaussian noise.
Abstract: The automatic frequency control (AFC) loop is beginning to play an important role in digital data links. There is very little published literature which deals with AFC loop implementations and performance in noise (in contrast to the large body of literature dealing with phase-locked loops). This paper presents a brief discussion of eight different AFC loop implementations, including the block diagram, discriminator characteristic, and loop tracking error in the presence of gaussian noise. Tracking performance of the various loops is compared in the presence of noise for CW, DPSK, and MFSK signaling.

Journal ArticleDOI
TL;DR: The results of a computer analysis of the distance properties of some of the best known rate 1/2, 1/3, and 1/4 codes to constraint length 14 include the truncated weight distributions of the codewords belonging to the incorrect subset which specifies the performance of the Viterbi algorithm.
Abstract: This paper reports the results of a computer analysis of the distance properties of some of the best known rate 1/2, 1/3, and 1/4 codes to constraint length 14. The data include the truncated weight distributions of the codewords belonging to the incorrect subset which specifies the performance of the Viterbi algorithm as well as the minimum asymptotic growth rate of the weights of unmerged codewords which has been conjectured to be related to the length of error events produced by Viterbi decoders.

Journal ArticleDOI
TL;DR: This work presents a new solution technique that is not doomed by the "statespace explosion" problem, and considers only the most probable states of the communication network, which can get upper and lower bounds and, hence, a good approximation of the network performance without having to analyze all possible states.
Abstract: In evaluating the performance of a communication network with unreliable components, researchers have traditionally approached the problem by enumerating all possible states of the system. Since the number of states of a communication network with n failure-prone components is 2nthese methods are restricted to small systems. We present a new solution technique that is not doomed by the "statespace explosion" problem. Instead of enumerating all possible fail states, we consider only the most probable states. Since the network operates in these states most of the time, we can get upper and lower bounds and, hence, a good approximation of the network performance without having to analyze all possible states. We illustrate our solution technique by analyzing network reliability, the expected number of communicating pairs, and network average delay for some particular networks.

Journal ArticleDOI
TL;DR: It is found that 16-PSK outperforms the two other modulation formats, and that the use of these codes can provide a substantial performance improvement even on a satellite channel.
Abstract: Currently, 4-PSK is the prevalent modulation format in use for digital satellite communications. To improve bandwidth efficiency, 8PSK could be used instead, but a higher power would be needed; to improve power efficiency, error-correcting codes could be used, but at the expense of a larger bandwidth. Recently, Ungerboeck [1] has proposed a class of codes in which a constellation of 2M signals is used to transmit information at the rate of log_{2} M bits per symbol, and has shown that coding gains of up to several decibels can be achieved on the additive white Gaussian noise (AWGN) channel with no increase in bandwidth occupancy and a relatively small added complexity. Thus, these codes seem to be particularly attractive for application in the band-limited environment typical of satellite communication systems, provided that the performance gain that they provide on the AWGN channel is not lost over a satellite channel. The goal of this work is to assess the performance of this class of codes when used to transmit 3 information bits per symbol on a band-limited, nonlinear satellite channel. Three modulation formats are considered, namely 16-PSK, 16-QAM, and a 16-ary amplitude-phase keying scheme with two amplitude levels. It is found that 16-PSK outperforms the two other modulation formats, and that the use of these codes can provide a substantial performance improvement even on a satellite channel.

Journal ArticleDOI
L. Miller, Jhong Lee, A. Kadrichu1
TL;DR: Results indicated that the higher number of hops per bit produced higher error probabilities as a result of increased combining losses when the square-law linear combining soft decision receiver is employed in demodulating the multihop-per-bit waveform.
Abstract: In this paper, error probability analyses are performed for a binary frequency-shift-keying (BFSK) system employing L hop/bit frequency-hopping (FH) spread-spectrum waveforms transmitted over a partial-band Gaussian noise jamming channel. The error probabilities for the L hop/bit BFSK/FH systems are obtained as the performance measure of the square-law linear combining soft decision receiver under the assumption of the worst-case partial-band jamming. The receiver in our analysis assumes no knowledge of jamming state (side information). Both exact and approximate (multiple bound-parameter Chernoff bound) solutions are obtained under two separate assumptions: with and without the system's thermal noise in the analyses. Numerical results of the error rates are graphically displayed as a function of signal-to-jamming power ratio with L and signal-to-noise ratio as parameters. All of our results, exact and approximate, indicated that the higher number of hops per bit produced higher error probabilities as a result of increased combining losses when the square-law linear combining soft decision receiver is employed in demodulating the multihop-per-bit waveform.

Journal ArticleDOI
TL;DR: It is shown by numerical examples that the simplest form of this approximation yields nearly optimal (asymptotic) performance for the problem of locally optimum detection.
Abstract: The Middleton Class A narrow-band non-Gaussian noise model [9]-[12] is examined. It is shown that this noise model (which is known to fit closely a variety of non-Gaussian noises) can itself be closely approximated by a computationally much simpler noise model. It is then shown by numerical examples that, for the problem of locally optimum detection, the simplest form of this approximation yields nearly optimal (asymptotic) performance. The performance of other simple suboptimal threshold detectors in Class A noise is also examined. Finally, a useful relationship between the Class A model and the e-mixture model is developed.

Journal ArticleDOI
TL;DR: The algorithms are based on Gallager's method and provide methods for iteratively updating the routing table entries of each node in a manner that guarantees convergence to a minimum delay routing and utilize second derivatives of the objective function.
Abstract: We propose a class of algorithms for finding an optimal quasi-static routing in a communication network. The algorithms are based on Gallager's method [1] and provide methods for iteratively updating the routing table entries of each node in a manner that guarantees convergence to a minimum delay routing. Their main feature is that they utilize second derivatives of the objective function and may be viewed as approximations to a constrained version of Newton's method. The use of second derivatives results in improved speed of convergence and automatic stepsize scaling with respect to level of traffic input. These advantages are of crucial importance for the practical implementation of the algorithm using distributed computation in an environment where input traffic statistics gradually change.

Journal ArticleDOI
TL;DR: The distribution of the output of the one-dimensional median filter is derived for several cases including the k th-order output distribution with any input distribution, which is then used in several illustrative examples of median filtering a signal plus white noise.
Abstract: The distribution of the output of the one-dimensional median filter is derived for several cases including the k th-order output distribution with any input distribution. This is then used in several illustrative examples of median filtering a signal plus white noise.

Journal ArticleDOI
Yu-Shuan Yeh1, S. Schwartz
TL;DR: The average outage probability is computed for centrally located base stations when multiple log-normal interferers are present and it is found that the outage probabilities for the two cases do not differ in a significant way.
Abstract: The mobile radio channel is characterized by three important factors: path losses larger than free space, fading typically taken as Rayleigh, and shadowing generally characterized as lognormal. For cellular systems, in order to determine acceptable reuse distances between base stations and to compare modulation methods, the probability of unacceptable cochannel interference (outage probability) has to be determined in the realistic situation where both fading and shadowing occur. In this paper, the average outage probability is computed for centrally located base stations when multiple log-normal interferers are present. This is done for both the mobile-to-base and base-to-mobile communication links. An unexpected result of this study is that the outage probabilities for the two cases do not differ in a significant way. Cumulative probability curves of the short-term average-signal-toaverage-interference ratio (SIR) are presented for a variety of system parameters: channel set number, propagation law exponent (γ), and dB spread (σ) of the log-normal distribution for the signal and interferers. An important observation is the large sensitivity of the performance curves to the propagation parameters: for a system with seven channel sets with a 10 dB SIR threshold, the average outage probability varies from 10 percent for \gamma = 3.7, \sigma = 6 dB, to 70 percent for \gamma = 3, \sigma = 14 dB. Alternatively, for a fixed outage objective of 10 percent, the required SIR threshold value ranges from -17 dB to 11 dB, depending on the propagation parameters. These variations make it imperative that accurate measurements of these parameters be obtained for the different service areas. Outage probabilities are also easily related to specific modulation methods and diversity approaches; detailed results are given for several representative cases.

Journal ArticleDOI
TL;DR: A new pilot tone SSB configuration, transparent tone-in-band (TTIB), which may be used in mobile radio systems from low-band VHF to microwave frequencies and a new technique utilizing TTIB is suggested to facilitate the use of coherent data systems.
Abstract: The paper describes a new pilot tone SSB configuration, transparent tone-in-band (TTIB), which may be used in mobile radio systems from low-band VHF to microwave frequencies. By utilizing audio signal processing techniques in the transmitter and receiver, the pilot reference tone may be positioned centrally within the RF channel bandwidth without losing the property of data transparency and also retains the many system advantages of tone-in-band SSB over the pilot carrier and tone-above-band schemes. Besides speech transmissions, results are presented for noncoherent FSK and DPSK data formats under white noise and Rayleigh fading conditions. Finally, a new technique utilizing TTIB is suggested to facilitate the use of coherent data systems.

Journal ArticleDOI
TL;DR: A relatively simple protocol is studied which is easy to implement and performs very well under a wide range of conditions and is shown to perform considerably better than the other go-back- N protocols, particularly in environments with a large number of receivers.
Abstract: In this paper we study some link control protocols for use in point-to-multipoint communication over broadcast links. We concentrate on automatic repeat request protocols of the go-back- N type and define, analyze, and compare three such protocols. A major contribution of this paper is a relatively simple protocol which is easy to implement and performs very well under a wide range of conditions. Our analytical models show that this protocol performs considerably better than the other go-back- N protocols, particularly in environments with a large number of receivers.

Journal ArticleDOI
TL;DR: In this paper, a test sequence generation algorithm for finite state machines is presented, where the tester or responder processes are forced to consider the timing of an interaction in which they have not taken part.
Abstract: Protocol testing for the purpose of certifying the implementation's adherence to the protocol specification can be done with a test architecture consisting of remote tester and local responder processes generating specific input stimuli, called test sequences, and observing the output produced by the implementation under test. It is possible to adapt test sequence generation techniques for finite state machines, such as transition tour, characterization, and checking sequence methods, to generate test sequences for protocols specified as incomplete finite state machines. For certain test sequences, the tester or responder processes are forced to consider the timing of an interaction in which they have not taken part; these test sequences are called nonsynchronizable. The three test sequence generation algorithms are modified to obtain synchronizable test sequences. The checking of a given protocol for intrinsic synchronization problems is also discussed. Complexities of synchronizable test sequence generation algorithms are given and complete testing of a protocol is shown to be infeasible. To extend the applicability of the characterization and checking sequences, different methods are proposed to enhance the protocol specifications: special test input interactions are defined and a methodology is developed to complete the protocol specifications.

Journal ArticleDOI
TL;DR: It is shown that the most common LLSE filter design can lead to performance inferior to that of various other filter designs, but results are also presented demonstrating that an LLSEfilter design motivated by the structure of the maximum-likelihood receiver leads to consistently superior performance.
Abstract: Linear least squares estimation (LLSE) techniques can provide an effective means of suppressing narrow-band interference in direct sequence (DS) spread-spectrum systems. In the results presented here, analytical expressions for bit error rate are derived for two DS spread-spectrum systems under the conditions of either tone or narrowband Gaussian interference. It is shown that the most common LLSE filter design can lead to performance inferior to that of various other filter designs. However, results are also presented demonstrating that an LLSE filter design motivated by the structure of the maximum-likelihood receiver leads to consistently superior performance. The performance of a system using this new design criterion is compared with that of an approximation to the maximum-likelihood (ML) receiver for the tone interference model and with that of the exact ML receiver for the Gaussian interference. Finally, it is shown that the bit error rate estimate obtained from application of a Gaussian approximation for the test statistic is overly pessimistic for the systems studied here.

Journal ArticleDOI
TL;DR: Several versions of a different technique which can result in better efficiency are shown which are not exact replicas of initially unacknowledged frames but are chosen to provide additional information to all sites having one or more nondecodable frames.
Abstract: A recent paper by Calo and Easton proposes a broadcast protocol for identical file transfers to M different sites wherein a large block of transmitted data is divided into N frames of B bits each, and in a second transmission cycle all frames not acknowledged by all sites are retransmitted. This paper shows several versions of a different technique which can result in better efficiency. In the technique, additional frames sent are not exact replicas of initially unacknowledged frames, but are chosen to provide additional information to all sites having one or more nondecodable frames. New frames are sent to provide additional information until all sites acknowledge the entire block.

Journal ArticleDOI
TL;DR: In this article, the authors developed a general formulation for MMSE equalization of interference in digital transmission diversity systems, which includes the use of available receiver decisions to assist in MMSE processing, and showed how the MMSE processor sacrifices diversity to suppress interference even when the interference arrives in the main beams of the receiver antenna patterns.
Abstract: Adaptive equalization is used in digital transmission systems with parallel fading channels. The equalization combines the diversity channels and reduces intersymbol interference due to multipath returns. When interference is present and correlated from channel to channel, the equalizer can also reduce its effect on the quality of information transfer, important applications for interference cancellation occur in diversity troposcatter systems in the presence of jamming, diversity high frequency (HF) systems which must cope with interfering skywaves, and space diversity line-of-sight (LOS) radio systems where adjacent channel interference is a problem. In this paper we develop the general formulation for minimum mean square error (MMSE) equalization of interference in digital transmission diversity systems. The problem formulation includes the use of available receiver decisions to assist in MMSE processing. The effects of intersymhol interference are included in the analysis through a critical approximation which assumes sufficient processor capability to reduce ISI effects to levels small enough for satisfactory communication. The analysis also develops he concept of additional implicit or intrinsic diversity which results from channel multipath dispersion. It shows how the MMSE processor sacrifices diversity to suppress interference even when the interference arrives in the main beams of the receiver antenna patterns. The condition of near synchronous same-path interference is also addressed. Because the spatial angle of arrival of the interference may result in delay differences between interference signals in different antenna channels, interference delay compensation may be required. We show that this effect is compensated for with a small number of appropriately spaced equalizer taps.

Journal ArticleDOI
TL;DR: In this article, the authors studied the application of group testing to the design of efficient algorithms for random multiple access communication systems and proposed and analyzed algorithms based on conventional and generalized group testing techniques.
Abstract: We study the application of group testing to the design of efficient algorithms for random multiple-access communication systems. Both direct transmission and reservation systems are considered for various types of channel feedback. We propose and analyze algorithms based on conventional and generalized group testing techniques. The proposed algorithms outperform TDMA and algorithms based on binary tree search and possess certain optimality properties.

Journal ArticleDOI
TL;DR: This research introduces extensions to the basic algorithm of [1] and offers two significant improvements: convergence speed is improved substantially, which means that the algorithms can better tolerate motion changes and diverse motion within the picture.
Abstract: A pel-recursive motion estimation technique for television coding was introduced by Netravali and Robbins [1]. This method basically involves computing a motion displacement and then separating the pels into predictable and unpredictable segments. The addresses required to identify these segments and thus enable the motion estimation algorithm are transmitted along with pel error data. The original algorithms are slow to converge, i.e., many iterations are required to obtain an accurate displacement estimate. This research introduces extensions to the basic algorithm of [1] and offers two significant improvements. First, convergence speed is improved substantially, which means that the algorithms can better tolerate motion changes and diverse motion within the picture. Secondly, explicit address information is not required, as it is contained implicitly in the motion compensation algorithm. For the sequences examined, this method proved superior to that of explicitly transmitting the address data. This algorithm has much better convergence rates and has prediction rates better than the original.

Journal ArticleDOI
TL;DR: This paper describes a general class of simple Viterbi detectors with reduced complexity compared to the optimum case, and finds the asymptotically optimum reduced-complexity receiver for a variety of transmitted schemes and various complexity reduction factors.
Abstract: Partial response continuous phase modulation (CPM) gives constant envelope digital modulation schemes with excellent power spectra. Both narrow main lobe and low spectral tails can be achieved. When these signals are detected in an optimum coherent maximum likelihood sequence detector (Viterbi detector), power efficient schemes can also be designed, sometimes at the expense of receiver complexity. This paper describes a general class of simple Viterbi detectors with reduced complexity compared to the optimum case. The key idea is that the approximate receiver is based on a less complex CPM scheme than the transmitted scheme. The asymptotically optimum reduced-complexity receiver is found for a variety of transmitted schemes and various complexity reduction factors, for a specific class of receivers and modulation indexes. A new distance measure is introduced for the performance analysis. Smooth schemes based on raised cosine pulses are analyzed and simulated for the case of simplified reception. A graceful performance degradation occurs with the reduction of complexity.

Journal ArticleDOI
TL;DR: The results show that the MDS codes are effective for both pure error detection and simultaneous error correction and detection.
Abstract: In this paper we investigate the performance of maximum-distance-separable codes with symbols from GF(q) when they are used for pure error detection or for simultaneous error correction and detection over a q -input and q -output discret memoryless channel with symbol error probability e. First we show that the probability of undetected error for an MDS code used for pure error detection is upper bounded by q^{-r} and decreases monotonically as edecreases from (q - 1)/q to 0, where r is the number of parity-check symbols of the code. Then we show that the probability of undetected error for an MDS code used for correcting t or fewer symbol errors is upper bounded by q^{-r} \Sum\min{i=0}\max{t}(\min{i} \max{n})(q - 1)^{i} and decreases monotonically as e decreases from (q - 1)/q to 0. These results show that the MDS codes are effective for both pure error detection and simultaneous error correction and detection.