The reliability function problems with fixed-length source coding for the general source are studied for all rates R. Our fundamental philosophy in doing so is to convert all of the reliability function problems to the pertinent computation problems in the large derivation-probability theory. It turns out that this kind of new methodology, which was previously developed by Han (see ibid., vol.43, p.1145-64, 1997), enables us to establish quite compact general formulas of the reliability function for general sources including all nonstationary and/or nonergodic sources with countably infinite alphabet. Such general formulas are presented from the information-spectrum point of view.
{"title":"The reliability functions of the general source with fixed-length coding","authors":"T. Han","doi":"10.1109/18.868482","DOIUrl":"https://doi.org/10.1109/18.868482","url":null,"abstract":"The reliability function problems with fixed-length source coding for the general source are studied for all rates R. Our fundamental philosophy in doing so is to convert all of the reliability function problems to the pertinent computation problems in the large derivation-probability theory. It turns out that this kind of new methodology, which was previously developed by Han (see ibid., vol.43, p.1145-64, 1997), enables us to establish quite compact general formulas of the reliability function for general sources including all nonstationary and/or nonergodic sources with countably infinite alphabet. Such general formulas are presented from the information-spectrum point of view.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"92 1","pages":"2117-2132"},"PeriodicalIF":0.0,"publicationDate":"2000-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85880513","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 depth distribution of a linear code was recently introduced by T. Etzion (see ibid., vol.43, pp.1361-3, July 1997). In this correspondence, a number of basic and interesting properties for the depth of finite words and the depth distribution of linear codes are obtained. In addition, we study the enumeration problem of counting the number of linear subcodes with the prescribed depth constraints, and derive some explicit and interesting enumeration formulas. Furthermore, we determine the depth distribution of Reed-Muller code RM (m,r). Finally, we show that there are exactly nine depth-equivalence classes for the ternary [11,6,5] Golay codes.
{"title":"On the depth distribution of linear codes","authors":"Luo Yuan, Fang-Wei Fu, V. Wei","doi":"10.1109/18.868491","DOIUrl":"https://doi.org/10.1109/18.868491","url":null,"abstract":"The depth distribution of a linear code was recently introduced by T. Etzion (see ibid., vol.43, pp.1361-3, July 1997). In this correspondence, a number of basic and interesting properties for the depth of finite words and the depth distribution of linear codes are obtained. In addition, we study the enumeration problem of counting the number of linear subcodes with the prescribed depth constraints, and derive some explicit and interesting enumeration formulas. Furthermore, we determine the depth distribution of Reed-Muller code RM (m,r). Finally, we show that there are exactly nine depth-equivalence classes for the ternary [11,6,5] Golay codes.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"151 1","pages":"2197-2203"},"PeriodicalIF":0.0,"publicationDate":"2000-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91128389","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 Z/sub 4/-linear Goethals-like code of length 2/sup m/ has 2/sup 2m+1-3m-2/ codewords and minimum Lee distance 8 for any odd integer m/spl ges/3. We present an algebraic decoding algorithm for all Z/sub 4/-linear Goethals-like codes C/sub k/ introduced by Helleseth et al.(1995, 1996). We use Dickson polynomials and their properties to solve the syndrome equations.
{"title":"On algebraic decoding of the Z4-linear Goethals-like codes","authors":"K. Ranto","doi":"10.1109/18.868490","DOIUrl":"https://doi.org/10.1109/18.868490","url":null,"abstract":"The Z/sub 4/-linear Goethals-like code of length 2/sup m/ has 2/sup 2m+1-3m-2/ codewords and minimum Lee distance 8 for any odd integer m/spl ges/3. We present an algebraic decoding algorithm for all Z/sub 4/-linear Goethals-like codes C/sub k/ introduced by Helleseth et al.(1995, 1996). We use Dickson polynomials and their properties to solve the syndrome equations.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"42 1","pages":"2193-2197"},"PeriodicalIF":0.0,"publicationDate":"2000-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82706020","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 obtain a lower bound on the linear complexity profile of the power generator of pseudo-random numbers modulo a Blum integer. A different method is also proposed to estimate the linear complexity profile of the Blum-Blum-Shub (1986) generator. In particular, these results imply that lattice reduction attacks on such generators are not feasible.
{"title":"On the linear complexity profile of the power generator","authors":"F. Griffin, I. Shparlinski","doi":"10.1109/18.868485","DOIUrl":"https://doi.org/10.1109/18.868485","url":null,"abstract":"We obtain a lower bound on the linear complexity profile of the power generator of pseudo-random numbers modulo a Blum integer. A different method is also proposed to estimate the linear complexity profile of the Blum-Blum-Shub (1986) generator. In particular, these results imply that lattice reduction attacks on such generators are not feasible.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"6 1","pages":"2159-2162"},"PeriodicalIF":0.0,"publicationDate":"2000-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75141529","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}
For pt. I see ibid., vol.46, p.755-88 (2000). The concept of context-free grammar (CFG)-based coding is extended to the case of countable-context models, yielding context-dependent grammar (CDG)-based coding. Given a countable-context model, a greedy CDG transform is proposed. Based on this greedy CDG transform, two universal lossless data compression algorithms, an improved sequential context-dependent algorithm and a hierarchical context-dependent algorithm, are then developed. It is shown that these algorithms are all universal in the sense that they can achieve asymptotically the entropy rate of any stationary, ergodic source with a finite alphabet. Moreover, it is proved that these algorithms' worst case redundancies among all individual sequences of length n from a finite alphabet are upper-bounded by d log log n/log n, as long as the number of distinct contexts grows with the sequence length n in the order of O(n/sup a/), where 0 < /spl alpha/ < 1 and d are positive constants. It is further shown that for some nonstationary sources, the proposed context-dependent algorithms can achieve better expected redundancies than any existing CFG-based codes, including the Lempel-Ziv (1978) algorithm, the multilevel pattern matching algorithm, and the context-free algorithms in Part I of this series of papers.
{"title":"Efficient universal lossless data compression algorithms based on a greedy sequential grammar transform .2. With context models","authors":"E. Yang, Dake He","doi":"10.1109/TIT.2003.818411","DOIUrl":"https://doi.org/10.1109/TIT.2003.818411","url":null,"abstract":"For pt. I see ibid., vol.46, p.755-88 (2000). The concept of context-free grammar (CFG)-based coding is extended to the case of countable-context models, yielding context-dependent grammar (CDG)-based coding. Given a countable-context model, a greedy CDG transform is proposed. Based on this greedy CDG transform, two universal lossless data compression algorithms, an improved sequential context-dependent algorithm and a hierarchical context-dependent algorithm, are then developed. It is shown that these algorithms are all universal in the sense that they can achieve asymptotically the entropy rate of any stationary, ergodic source with a finite alphabet. Moreover, it is proved that these algorithms' worst case redundancies among all individual sequences of length n from a finite alphabet are upper-bounded by d log log n/log n, as long as the number of distinct contexts grows with the sequence length n in the order of O(n/sup a/), where 0 < /spl alpha/ < 1 and d are positive constants. It is further shown that for some nonstationary sources, the proposed context-dependent algorithms can achieve better expected redundancies than any existing CFG-based codes, including the Lempel-Ziv (1978) algorithm, the multilevel pattern matching algorithm, and the context-free algorithms in Part I of this series of papers.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"87 1","pages":"2874-2894"},"PeriodicalIF":0.0,"publicationDate":"2000-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83514072","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}
Ying Guo, P. Bartlett, J. Shawe-Taylor, R. C. Williamson
Support vector (SV) machines are linear classifiers that use the maximum margin hyperplane in a feature space defined by a kernel function. Until recently, the only bounds on the generalization performance of SV machines (within Valiant's probably approximately correct framework) took no account of the kernel used except in its effect on the margin and radius. More recently, it has been shown that one can bound the relevant covering numbers using tools from functional analysis. In this paper, we show that the resulting bound can be greatly simplified. The new bound involves the eigenvalues of the integral operator induced by the kernel. It shows that the effective dimension depends on the rate of decay of these eigenvalues. We present an explicit calculation of covering numbers for an SV machine using a Gaussian kernel, which is significantly better than that implied by previous results.
{"title":"Covering numbers for support vector machines","authors":"Ying Guo, P. Bartlett, J. Shawe-Taylor, R. C. Williamson","doi":"10.1145/307400.307467","DOIUrl":"https://doi.org/10.1145/307400.307467","url":null,"abstract":"Support vector (SV) machines are linear classifiers that use the maximum margin hyperplane in a feature space defined by a kernel function. Until recently, the only bounds on the generalization performance of SV machines (within Valiant's probably approximately correct framework) took no account of the kernel used except in its effect on the margin and radius. More recently, it has been shown that one can bound the relevant covering numbers using tools from functional analysis. In this paper, we show that the resulting bound can be greatly simplified. The new bound involves the eigenvalues of the integral operator induced by the kernel. It shows that the effective dimension depends on the rate of decay of these eigenvalues. We present an explicit calculation of covering numbers for an SV machine using a Gaussian kernel, which is significantly better than that implied by previous results.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"112 1","pages":"239-250"},"PeriodicalIF":0.0,"publicationDate":"1999-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73367560","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}
Pub Date : 1999-06-24DOI: 10.1142/9789812563071_0003
T. Ogawa, H. Nagaoka
The hypothesis testing problem for two quantum states is treated. We show a new inequality between the errors of the first kind and the second kind, which complements the result of Hiai and Petz (1991) to establish the quantum version of Stein's lemma. The inequality is also used to show a bound on the probability of errors of the first kind when the power exponent for the probability of errors of the second kind exceeds the quantum relative entropy, which yields the strong converse in quantum hypothesis testing. Finally, we discuss the relation between the bound and the power exponent derived by Han and Kobayashi (1989) in classical hypothesis testing.
{"title":"Strong converse and Stein's lemma in quantum hypothesis testing","authors":"T. Ogawa, H. Nagaoka","doi":"10.1142/9789812563071_0003","DOIUrl":"https://doi.org/10.1142/9789812563071_0003","url":null,"abstract":"The hypothesis testing problem for two quantum states is treated. We show a new inequality between the errors of the first kind and the second kind, which complements the result of Hiai and Petz (1991) to establish the quantum version of Stein's lemma. The inequality is also used to show a bound on the probability of errors of the first kind when the power exponent for the probability of errors of the second kind exceeds the quantum relative entropy, which yields the strong converse in quantum hypothesis testing. Finally, we discuss the relation between the bound and the power exponent derived by Han and Kobayashi (1989) in classical hypothesis testing.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"25 1","pages":"2428-2433"},"PeriodicalIF":0.0,"publicationDate":"1999-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79675614","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 fast algorithm for the computation of the /spl rho/-representation of n-dimensional discrete Fourier transform (DFT) is given, where /spl rho/ is an mth primitive root of unity. Applying this algorithm to the standard /spl rho/-representation of the DFT of /spl rho//sup f(x)/, the best linear approximation of a function f(x) can be easily obtained when the codomain of f(x) is Z/sub m/. A spectral characterization of correlation-immune functions over Z, is also presented in terms of the DFT of /spl zeta//sup f(x)/.
{"title":"Best Linear Approximation and Correlation Immunity of Functions Over Z*m","authors":"Jinjun Zhou, Weihong Chen, Fengxiu Gao","doi":"10.1109/18.746825","DOIUrl":"https://doi.org/10.1109/18.746825","url":null,"abstract":"A fast algorithm for the computation of the /spl rho/-representation of n-dimensional discrete Fourier transform (DFT) is given, where /spl rho/ is an mth primitive root of unity. Applying this algorithm to the standard /spl rho/-representation of the DFT of /spl rho//sup f(x)/, the best linear approximation of a function f(x) can be easily obtained when the codomain of f(x) is Z/sub m/. A spectral characterization of correlation-immune functions over Z, is also presented in terms of the DFT of /spl zeta//sup f(x)/.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"12 1","pages":"303-308"},"PeriodicalIF":0.0,"publicationDate":"1999-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75992143","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 present new formulas for solving quadratic equations with distinct roots in certain finite fields. We develop in detail formulas for solving such quadratics over GF (4), GF(16), and GF (256) and the approach we take is applicable for solving quadratics over GF (2k), with k a power of 2.
{"title":"New Formulas for Solving Quadratic Equations over Certain Finite Fields","authors":"C. Walker","doi":"10.1109/18.746816","DOIUrl":"https://doi.org/10.1109/18.746816","url":null,"abstract":"We present new formulas for solving quadratic equations with distinct roots in certain finite fields. We develop in detail formulas for solving such quadratics over GF (4), GF(16), and GF (256) and the approach we take is applicable for solving quadratics over GF (2k), with k a power of 2.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"80 1","pages":"283-284"},"PeriodicalIF":0.0,"publicationDate":"1999-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79331524","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 find the minimum distance of the nonextended [83,42] ternary quadratic residue code to be 20.
我们发现非扩展的[83,42]三元二次剩余码的最小距离为20。
{"title":"The Minimum Distance of the [83, 42] Ternary Quadratic Residue Code","authors":"Doug Kuhlman","doi":"10.1109/18.746814","DOIUrl":"https://doi.org/10.1109/18.746814","url":null,"abstract":"We find the minimum distance of the nonextended [83,42] ternary quadratic residue code to be 20.","PeriodicalId":13250,"journal":{"name":"IEEE Trans. Inf. Theory","volume":"13 1","pages":"282"},"PeriodicalIF":0.0,"publicationDate":"1999-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84283222","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：