Sixteen new binary quasi-cyclic linear codes improving the best known lower bounds on minimum distance in Brouwer's tables are constructed. The parameters of these codes are [102, 26, 32], [102, 27, 30], [142, 35, 40], [142, 36, 38] [146, 36, 40], [170, 16, 72], [170, 20, 66], [170, 33, 52] [170, 36, 50], [178, 33, 56], [178, 34, 54], [182, 27, 64] [182, 36, 56], [186, 17, 76], [210, 23, 80], [254, 22, 102] Sixty cyclic and thirty quasi-cyclic codes, which attain the respective bounds in Brouwer's table and are not included in Chen's table are presented as well.
{"title":"New binary one-generator quasi-cyclic codes","authors":"R. Daskalov, P. Hristov","doi":"10.1109/TIT.2003.819337","DOIUrl":"https://doi.org/10.1109/TIT.2003.819337","url":null,"abstract":"Sixteen new binary quasi-cyclic linear codes improving the best known lower bounds on minimum distance in Brouwer's tables are constructed. The parameters of these codes are [102, 26, 32], [102, 27, 30], [142, 35, 40], [142, 36, 38] [146, 36, 40], [170, 16, 72], [170, 20, 66], [170, 33, 52] [170, 36, 50], [178, 33, 56], [178, 34, 54], [182, 27, 64] [182, 36, 56], [186, 17, 76], [210, 23, 80], [254, 22, 102] Sixty cyclic and thirty quasi-cyclic codes, which attain the respective bounds in Brouwer's table and are not included in Chen's table are presented as well.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"12 1","pages":"3001-3005"},"PeriodicalIF":0.0,"publicationDate":"2003-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88261723","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The rank distance was introduced by E.M. Gabidulin (see Probl. Pered. Inform., vol.21, p.1-12, 1985). He determined an upper bound for the minimum rank distance of a code. Moreover, he constructed a class of codes which meet this bound: the so-called Gabidulin codes. We first characterize the linear and semilinear isometries for the rank distance. Then we determine the isometry group and the permutation group of Gabidulin codes of any length. We give a characterization of equivalent Gabidulin codes. Finally, we prove that the number of equivalence classes of Gabidulin codes is exactly the number of equivalence classes of vector spaces of dimension n contained in GF(p/sup m/) under some particular relations.
{"title":"Isometries for rank distance and permutation group of Gabidulin codes","authors":"T. Berger","doi":"10.1109/TIT.2003.819322","DOIUrl":"https://doi.org/10.1109/TIT.2003.819322","url":null,"abstract":"The rank distance was introduced by E.M. Gabidulin (see Probl. Pered. Inform., vol.21, p.1-12, 1985). He determined an upper bound for the minimum rank distance of a code. Moreover, he constructed a class of codes which meet this bound: the so-called Gabidulin codes. We first characterize the linear and semilinear isometries for the rank distance. Then we determine the isometry group and the permutation group of Gabidulin codes of any length. We give a characterization of equivalent Gabidulin codes. Finally, we prove that the number of equivalence classes of Gabidulin codes is exactly the number of equivalence classes of vector spaces of dimension n contained in GF(p/sup m/) under some particular relations.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"77 1","pages":"3016-3019"},"PeriodicalIF":0.0,"publicationDate":"2003-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72637144","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A new coding construction scheme of block codes using short base codes and permutations that enables the construction of binary self-dual codes is presented in Cadic et al. (2001) and Carlach et al. (1999, 2000). The scheme leads to doubly-even (resp,. singly-even) self-dual codes provided the base code is a doubly-even self-dual code and the number of permutations is even (resp., odd). We study the particular case where the base code is the [8, 4, 4] extended Hamming. In this special case, we construct a new [88, 44, 16] extremal doubly-even self-dual code and we give a new unified construction of the five [32, 16, 8] extremal doubly-even self-dual codes.
Cadic et al.(2001)和carach et al.(1999,2000)提出了一种新的利用短基码和置换的分组码编码构造方案，该方案能够构造二进制自对偶码。该方案可实现双平衡。假设基码是双偶自对偶码，并且排列的数目是偶的(例如:奇怪的)。我们研究了基码为[8,4,4]扩展汉明的特殊情况。在这种特殊情况下，构造了一个新的[88,44,16]极值双偶自对偶码，并给出了5个[32,16,8]极值双偶自对偶码的一个新的统一构造。
{"title":"A systematic construction of self-dual codes","authors":"J. Carlach, A. Otmani","doi":"10.1109/TIT.2003.815814","DOIUrl":"https://doi.org/10.1109/TIT.2003.815814","url":null,"abstract":"A new coding construction scheme of block codes using short base codes and permutations that enables the construction of binary self-dual codes is presented in Cadic et al. (2001) and Carlach et al. (1999, 2000). The scheme leads to doubly-even (resp,. singly-even) self-dual codes provided the base code is a doubly-even self-dual code and the number of permutations is even (resp., odd). We study the particular case where the base code is the [8, 4, 4] extended Hamming. In this special case, we construct a new [88, 44, 16] extremal doubly-even self-dual code and we give a new unified construction of the five [32, 16, 8] extremal doubly-even self-dual codes.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"13 1","pages":"3005-3009"},"PeriodicalIF":0.0,"publicationDate":"2003-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74783416","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Chase decoders permit flexible use of reliability information in algebraic decoding algorithms for error-correcting block codes of Hamming distance d. The least complex version of the original Chase algorithms uses roughly d/2 trials of a conventional binary decoder, after which the best decoding result is selected as the final output. On certain channels, this approach achieves asymptotically the same performance as maximum-likelihood (ML) decoding. In this correspondence, the performance of Chase-like decoders with even less trials is studied. Most strikingly, it turns out that asymptotically optimal performance can be achieved by a version which uses only about d/4 trials.
{"title":"Limited-trial Chase decoding","authors":"G. Arico, J. Weber","doi":"10.1109/TIT.2003.818397","DOIUrl":"https://doi.org/10.1109/TIT.2003.818397","url":null,"abstract":"Chase decoders permit flexible use of reliability information in algebraic decoding algorithms for error-correcting block codes of Hamming distance d. The least complex version of the original Chase algorithms uses roughly d/2 trials of a conventional binary decoder, after which the best decoding result is selected as the final output. On certain channels, this approach achieves asymptotically the same performance as maximum-likelihood (ML) decoding. In this correspondence, the performance of Chase-like decoders with even less trials is studied. Most strikingly, it turns out that asymptotically optimal performance can be achieved by a version which uses only about d/4 trials.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"42 9 1","pages":"2972-2975"},"PeriodicalIF":0.0,"publicationDate":"2003-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72985122","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Orthogonal space-time block coding proposed recently by Alamouti (1998) and Tarokh et al. (1999) is a promising scheme for information transmission over Rayleigh-fading channels using multiple transmit antennas due to its favorable characteristics of having full transmit diversity and a decoupled maximum-likelihood (ML) decoding algorithm. Tarokh et al. extended the theory of classical orthogonal designs to the theory of generalized, real, or complex, linear processing orthogonal designs and then applied the theory of generalized orthogonal designs to construct space-time block codes (STBC) with the maximum possible diversity order while having a simple decoding algorithm for any given number of transmit and receive antennas. It has been known that the STBC constructed in this way can achieve the maximum possible rate of one for every number of transmit antennas using any arbitrary real constellation and for two transmit antennas using any arbitrary complex constellation. Contrary to this, in this correspondence we prove that there does not exist rate-one STBC from generalized complex linear processing orthogonal designs for more than two transmit antennas using any arbitrary complex constellation.
{"title":"On the nonexistence of rate-one generalized complex orthogonal designs","authors":"Xue-Bin Liang, X. Xia","doi":"10.1109/TIT.2003.818396","DOIUrl":"https://doi.org/10.1109/TIT.2003.818396","url":null,"abstract":"Orthogonal space-time block coding proposed recently by Alamouti (1998) and Tarokh et al. (1999) is a promising scheme for information transmission over Rayleigh-fading channels using multiple transmit antennas due to its favorable characteristics of having full transmit diversity and a decoupled maximum-likelihood (ML) decoding algorithm. Tarokh et al. extended the theory of classical orthogonal designs to the theory of generalized, real, or complex, linear processing orthogonal designs and then applied the theory of generalized orthogonal designs to construct space-time block codes (STBC) with the maximum possible diversity order while having a simple decoding algorithm for any given number of transmit and receive antennas. It has been known that the STBC constructed in this way can achieve the maximum possible rate of one for every number of transmit antennas using any arbitrary real constellation and for two transmit antennas using any arbitrary complex constellation. Contrary to this, in this correspondence we prove that there does not exist rate-one STBC from generalized complex linear processing orthogonal designs for more than two transmit antennas using any arbitrary complex constellation.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"36 1","pages":"2984-2988"},"PeriodicalIF":0.0,"publicationDate":"2003-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81121466","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we propose techniques for the construction of frequency-coding sequences that give rise to frequency-coded waveforms having ambiguity functions with a clear area - containing no sidelobes - in a connected region surrounding the main lobe. These constructed sequences are called pushing sequences. First, two important properties of pushing sequences are investigated: the group D/sub 4/ dihedral symmetry property and the frequency omission property. Using the group D/sub 4/ dihedral symmetry property, we show how to construct additional pushing sequences from a given pushing sequence. Using the frequency omission property, we show how to construct pushing sequences of any length N and design proper frequency-coded waveforms that meet specific constraints in the frequency domain. Next, we use the Lempel T/sub 4/ construction of Costas sequences to construct pushing sequences with power 1. Finally, we show how to construct pushing sequences with any desired power using Lee codewords. Because these arbitrary-power pushing sequences constructed using Lee codewords do not have the Costas property, we derive expressions for the pattern of hits in the geometric array. Based on this, the general form of the positions and levels of all the sidelobe peaks are derived.
{"title":"Frequency-coded waveforms for enhanced delay-Doppler resolution","authors":"Chieh-Fu Chang, M. Bell","doi":"10.1109/TIT.2003.818408","DOIUrl":"https://doi.org/10.1109/TIT.2003.818408","url":null,"abstract":"In this paper, we propose techniques for the construction of frequency-coding sequences that give rise to frequency-coded waveforms having ambiguity functions with a clear area - containing no sidelobes - in a connected region surrounding the main lobe. These constructed sequences are called pushing sequences. First, two important properties of pushing sequences are investigated: the group D/sub 4/ dihedral symmetry property and the frequency omission property. Using the group D/sub 4/ dihedral symmetry property, we show how to construct additional pushing sequences from a given pushing sequence. Using the frequency omission property, we show how to construct pushing sequences of any length N and design proper frequency-coded waveforms that meet specific constraints in the frequency domain. Next, we use the Lempel T/sub 4/ construction of Costas sequences to construct pushing sequences with power 1. Finally, we show how to construct pushing sequences with any desired power using Lee codewords. Because these arbitrary-power pushing sequences constructed using Lee codewords do not have the Costas property, we derive expressions for the pattern of hits in the geometric array. Based on this, the general form of the positions and levels of all the sidelobe peaks are derived.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"735 ","pages":"2960-2971"},"PeriodicalIF":0.0,"publicationDate":"2003-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91463859","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the estimation of the multivariate probability density function f(x/sub 1/,...,x/sub p/) of X/sub 1/,...,X/sub p/ of a stationary positively or negatively associated (PA or NA) random process {X/sub i/}/sub i=1//sup /spl infin// from noisy observations. Both ordinary smooth and super smooth noise are considered. Quadratic mean and asymptotic normality results are established.
{"title":"Deconvolving multivariate kernel density estimates from contaminated associated observations","authors":"E. Masry","doi":"10.1109/TIT.2003.818415","DOIUrl":"https://doi.org/10.1109/TIT.2003.818415","url":null,"abstract":"We consider the estimation of the multivariate probability density function f(x/sub 1/,...,x/sub p/) of X/sub 1/,...,X/sub p/ of a stationary positively or negatively associated (PA or NA) random process {X/sub i/}/sub i=1//sup /spl infin// from noisy observations. Both ordinary smooth and super smooth noise are considered. Quadratic mean and asymptotic normality results are established.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"1 1","pages":"2941-2952"},"PeriodicalIF":0.0,"publicationDate":"2003-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79719572","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the problem of lossy joint source-channel coding in a communication system where the encoder has access to channel state information (CSI) and the decoder has access to side information that is correlated to the source. This configuration combines the Wyner-Ziv (1976) model of pure lossy source coding with side information at the decoder and the Shannon/Gel'fand-Pinsker (1958, 1980) model of pure channel coding with CSI at the encoder. We prove a separation theorem for this communication system, which asserts that there is no loss in asymptotic optimality in applying, first, an optimal Wyner-Ziv source code and, then, an optimal Gel'fand-Pinsker channel code. We then derive conditions for the optimality of a symbol-by-symbol (scalar) source-channel code, and demonstrate situations where these conditions are met. Finally, we discuss a few practical applications, including overlaid communication where the model under discussion is useful.
{"title":"On joint source-channel coding for the Wyner-Ziv source and the Gel'fand-Pinsker channel","authors":"N. Merhav, S. Shamai","doi":"10.1109/TIT.2003.818410","DOIUrl":"https://doi.org/10.1109/TIT.2003.818410","url":null,"abstract":"We consider the problem of lossy joint source-channel coding in a communication system where the encoder has access to channel state information (CSI) and the decoder has access to side information that is correlated to the source. This configuration combines the Wyner-Ziv (1976) model of pure lossy source coding with side information at the decoder and the Shannon/Gel'fand-Pinsker (1958, 1980) model of pure channel coding with CSI at the encoder. We prove a separation theorem for this communication system, which asserts that there is no loss in asymptotic optimality in applying, first, an optimal Wyner-Ziv source code and, then, an optimal Gel'fand-Pinsker channel code. We then derive conditions for the optimality of a symbol-by-symbol (scalar) source-channel code, and demonstrate situations where these conditions are met. Finally, we discuss a few practical applications, including overlaid communication where the model under discussion is useful.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"9 1","pages":"2844-2855"},"PeriodicalIF":0.0,"publicationDate":"2003-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87689779","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study a multiuser detection system for code-division multiple access (CDMA). We show that applying multistage hard-decision parallel interference cancellation (HD-PIC) significantly improves performance compared to the matched filter system. In (multistage) HD-PIC, estimates of the interfering signals are used iteratively to improve knowledge of the desired signal. We use large deviation theory to show that the bit-error probability (BEP) is exponentially small when the number of users is fixed and the processing gain increases. We investigate the exponential rate of the BEP after several stages of HD-PIC. We propose to use the exponential rate of the BEP as a measure of performance, rather than the signal-to-noise ratio (SNR), which is often not reliable in multiuser detection models when the system is lightly loaded. We show that the exponential rate of the BEP remains fixed after a finite number of stages, resulting in an optimal hard-decision system. When the number of users becomes large, the exponential rate of the BEP converges to (log 2)/2 $1/4. We provide guidelines for the number of stages necessary to obtain this asymptotic exponential rate. We also give Chernoff bounds on the BEPs. These estimates show that the BEPs are quite small as long as k = o(n/log n) when the number of stages of HD-PIC is fixed, and even exponentially small when k = O(n) for the optimal HD-PIC system, and where k is the number of users in the system and n is the processing gain. Finally, we extend the results to the case where the number of stages depends on k in a certain manner. The above results are proved for a noiseless channel, and we argue that we expect similar results in a noisy channel as long as the two-sided spectrum of the noise decreases proportionally to n.
{"title":"Performance of DS-CDMA systems with optimal hard-decision parallel interference cancellation","authors":"R. Hofstad, M. J. Klok","doi":"10.1109/TIT.2003.818413","DOIUrl":"https://doi.org/10.1109/TIT.2003.818413","url":null,"abstract":"We study a multiuser detection system for code-division multiple access (CDMA). We show that applying multistage hard-decision parallel interference cancellation (HD-PIC) significantly improves performance compared to the matched filter system. In (multistage) HD-PIC, estimates of the interfering signals are used iteratively to improve knowledge of the desired signal. We use large deviation theory to show that the bit-error probability (BEP) is exponentially small when the number of users is fixed and the processing gain increases. We investigate the exponential rate of the BEP after several stages of HD-PIC. We propose to use the exponential rate of the BEP as a measure of performance, rather than the signal-to-noise ratio (SNR), which is often not reliable in multiuser detection models when the system is lightly loaded. We show that the exponential rate of the BEP remains fixed after a finite number of stages, resulting in an optimal hard-decision system. When the number of users becomes large, the exponential rate of the BEP converges to (log 2)/2 $1/4. We provide guidelines for the number of stages necessary to obtain this asymptotic exponential rate. We also give Chernoff bounds on the BEPs. These estimates show that the BEPs are quite small as long as k = o(n/log n) when the number of stages of HD-PIC is fixed, and even exponentially small when k = O(n) for the optimal HD-PIC system, and where k is the number of users in the system and n is the processing gain. Finally, we extend the results to the case where the number of stages depends on k in a certain manner. The above results are proved for a noiseless channel, and we argue that we expect similar results in a noisy channel as long as the two-sided spectrum of the noise decreases proportionally to n.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"3 1","pages":"2918-2940"},"PeriodicalIF":0.0,"publicationDate":"2003-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89305079","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An entropy-constrained quantizer Q is optimal if it minimizes the expected distortion D(Q) subject to a constraint on the output entropy H(Q). We use the Lagrangian formulation to show the existence and study the structure of optimal entropy-constrained quantizers that achieve a point on the lower convex hull of the operational distortion-rate function D/sub h/(R) = inf/sub Q/{D(Q) : H(Q) /spl les/ R}. In general, an optimal entropy-constrained quantizer may have a countably infinite number of codewords. Our main results show that if the tail of the source distribution is sufficiently light (resp., heavy) with respect to the distortion measure, the Lagrangian-optimal entropy-constrained quantizer has a finite (resp., infinite) number of codewords. In particular, for the squared error distortion measure, if the tail of the source distribution is lighter than the tail of a Gaussian distribution, then the Lagrangian-optimal quantizer has only a finite number of codewords, while if the tail is heavier than that of the Gaussian, the Lagrangian-optimal quantizer has an infinite number of codewords.
熵约束量化器Q是最优的，如果它在输出熵H(Q)的约束下最小化预期失真D(Q)。我们利用拉格朗日公式证明了最优熵约束量化器的存在性，并研究了最优熵约束量化器的结构，这些量化器实现了操作失真率函数D/sub h/(R) = inf/sub Q/{D(Q): h (Q) /spl les/ R}的下凸包上的一个点。一般来说，一个最优的熵约束量化器可以有无数个码字。我们的主要结果表明，如果光源分布的尾部足够轻(例如:相对于失真度量，拉格朗日最优熵约束量化器具有有限的响应。(无限)码字数。特别是对于平方误差失真度量，如果源分布的尾部比高斯分布的尾部轻，则拉格朗日最优量化器只有有限个码字，而如果源分布的尾部比高斯分布的尾部重，则拉格朗日最优量化器有无限个码字。
{"title":"Do optimal entropy-constrained quantizers have a finite or infinite number of codewords?","authors":"A. György, T. Linder, P. Chou, B. J. Betts","doi":"10.1109/TIT.2003.819340","DOIUrl":"https://doi.org/10.1109/TIT.2003.819340","url":null,"abstract":"An entropy-constrained quantizer Q is optimal if it minimizes the expected distortion D(Q) subject to a constraint on the output entropy H(Q). We use the Lagrangian formulation to show the existence and study the structure of optimal entropy-constrained quantizers that achieve a point on the lower convex hull of the operational distortion-rate function D/sub h/(R) = inf/sub Q/{D(Q) : H(Q) /spl les/ R}. In general, an optimal entropy-constrained quantizer may have a countably infinite number of codewords. Our main results show that if the tail of the source distribution is sufficiently light (resp., heavy) with respect to the distortion measure, the Lagrangian-optimal entropy-constrained quantizer has a finite (resp., infinite) number of codewords. In particular, for the squared error distortion measure, if the tail of the source distribution is lighter than the tail of a Gaussian distribution, then the Lagrangian-optimal quantizer has only a finite number of codewords, while if the tail is heavier than that of the Gaussian, the Lagrangian-optimal quantizer has an infinite number of codewords.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"29 1","pages":"3031-3037"},"PeriodicalIF":0.0,"publicationDate":"2003-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87520755","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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