Refinement of the Sphere-Packing Bound: Asymmetric Channels

We provide a refinement of the sphere-packing bound for constant composition codes over asymmetric discrete memoryless channels that improves the subexponential factor in front of the exponent. The order of our subexponential factor is Ω(N<sup>-0.5(1+ε+ρ</sup><sup>R</sup><sup>*</sup>)) for any ϵ > 0, where ρ<sub>R</sub>* is the left derivative of the sphere-packing exponent at rate R and N is the blocklength.

[1]  J. Wolfowitz.  The coding of messages subject to chance errors , 1957 .

[2]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[3]  D. A. Bell,et al.  Information Theory and Reliable Communication , 1969 .

[4]  Elwyn R. Berlekamp,et al.  Lower Bounds to Error Probability for Coding on Discrete Memoryless Channels. II , 1967, Inf. Control..

[5]  Robert G. Gallager,et al.  A simple derivation of the coding theorem and some applications , 1965, IEEE Trans. Inf. Theory.

[6]  Meir Feder,et al.  Finite-Dimensional Infinite Constellations , 2011, IEEE Transactions on Information Theory.

[7]  R. Rockafellar.  Convex Analysis: (pms-28) , 1970 .

[8]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[9]  N. Sloane,et al.  Lower Bounds to Error Probability for Coding on Discrete Memoryless Channels. I , 1993 .

[10]  Aaron B. Wagner,et al.  Moderate deviation analysis of channel coding: Discrete memoryless case , 2010, 2010 IEEE International Symposium on Information Theory.

[11]  H. Vincent Poor,et al.  Feedback in the Non-Asymptotic Regime , 2011, IEEE Transactions on Information Theory.

[12]  Richard E. Blahut,et al.  Hypothesis testing and information theory , 1974, IEEE Trans. Inf. Theory.

[13]  Donald E. Knuth,et al.  Big Omicron and big Omega and big Theta , 1976, SIGA.

[14]  S. Verdú,et al.  Channel dispersion and moderate deviations limits for memoryless channels , 2010, 2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[15]  Masahito Hayashi,et al.  Information Spectrum Approach to Second-Order Coding Rate in Channel Coding , 2008, IEEE Transactions on Information Theory.

[16]  Evgueni Haroutunian,et al.  Reliability Criteria in Information Theory and in Statistical Hypothesis Testing , 2008, Found. Trends Commun. Inf. Theory.

[17]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[18]  Igal Sason,et al.  An Improved Sphere-Packing Bound for Finite-Length Codes Over Symmetric Memoryless Channels , 2006, IEEE Transactions on Information Theory.

[19]  G. A. Barnard,et al.  Transmission of Information: A Statistical Theory of Communications. , 1961 .

[20]  Mill Johannes G.A. Van,et al.  Transmission Of Information , 1961 .

[21]  H. Vincent Poor,et al.  Channel Coding Rate in the Finite Blocklength Regime , 2010, IEEE Transactions on Information Theory.

[22]  R. Durrett.  Probability: Theory and Examples , 1993 .

[23]  Aaron B. Wagner,et al.  Refinement of the sphere-packing bound , 2012, 2012 IEEE International Symposium on Information Theory Proceedings.

[24]  Irina Gennad'evna Shevtsova,et al.  A NEW MOMENT-TYPE ESTIMATE OF CONVERGENCE RATE IN THE LYAPUNOV THEOREM ∗ , 2011 .

[25]  R. R. Bahadur,et al.  On Deviations of the Sample Mean , 1960 .

[26]  Vincent Y. F. Tan,et al.  Moderate-deviations of lossy source coding for discrete and Gaussian sources , 2011, 2012 IEEE International Symposium on Information Theory Proceedings.

[27]  Igal Sason,et al.  Moderate deviations analysis of binary hypothesis testing , 2011, 2012 IEEE International Symposium on Information Theory Proceedings.

[28]  A. Banerjee.  Convex Analysis and Optimization , 2006 .

[29]  Amir Dembo,et al.  Large Deviations Techniques and Applications , 1998 .

[30]  Aaron B. Wagner Member.  Refinement of the Random Coding Bound , 2012 .

[31]  Aaron B. Wagner,et al.  Refinement of the Random Coding Bound , 2012, IEEE Transactions on Information Theory.

[32]  Ying Xiong.  Nonlinear Optimization , 2014 .

[33]  C. Esseen.  Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law , 1945 .

[34]  N. R. Chaganty,et al.  Strong Large Deviation and Local Limit Theorems , 1993 .

[35]  Aaron B. Wagner,et al.  Refinement of the sphere packing bound for symmetric channels , 2011, 2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[36]  Marc P. C. Fossorier,et al.  Sphere-packing bounds revisited for moderate block lengths , 2004, IEEE Transactions on Information Theory.

[37]  W. Fenchel.  On Conjugate Convex Functions , 1949, Canadian Journal of Mathematics.

[38]  Imre Csiszár,et al.  Information Theory - Coding Theorems for Discrete Memoryless Systems, Second Edition , 2011 .

[39]  R. Rockafellar.  Minimax Theorems and Conjugate Saddle-Functions. , 1964 .

[40]  A. Ruszczynski,et al.  Nonlinear Optimization , 2006 .

[41]  Richard E. Blahut,et al.  Principles and practice of information theory , 1987 .