Refinement of the Random Coding Bound

An improved pre-factor for the random coding bound is proved. Specifically, for positive dispersion channels, if a certain regularity condition is satisfied, then for any ϵ > 0 a pre-factor of O(N-1/2(1-ϵ+ρ*w(R))) is achievable for rates above the critical rate, where ρ*w (R) is related to the slope of the random coding bound at rate R. If the regularity condition is not satisfied, then a pre-factor of O(N-1/2) is achievable. Recent work of the authors shows that the former case is nearly the best possible.

[1]  P. Ney.  Dominating Points and the Asymptotics of Large Deviations for Random Walk on $\mathbb{R}^d$ , 1983 .

[2]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[3]  Claude E. Shannon,et al.  Certain Results in Coding Theory for Noisy Channels , 1957, Inf. Control..

[4]  Robert G. Gallager,et al.  The random coding bound is tight for the average code (Corresp.) , 1973, IEEE Trans. Inf. Theory.

[5]  H. Vincent Poor,et al.  Channel coding: non-asymptotic fundamental limits , 2010 .

[6]  Aaron B. Wagner,et al.  Moderate Deviations in Channel Coding , 2012, IEEE Transactions on Information Theory.

[7]  Aaron B. Wagner,et al.  Refinement of the random coding bound , 2012 .

[8]  Brian L. Hughes,et al.  On the error probability of signals in additive white Gaussian noise , 1991, IEEE Trans. Inf. Theory.

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

[10]  C. Esseen.  On the concentration function of a sum of independent random variables , 1968 .

[11]  W. H. Young.  On Multiple Integration by Parts and the Second Theorem of the Mean , 1917 .

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

[13]  Kamil Sh. Zigangirov,et al.  Tighter bounds on the error probability of fixed convolutional codes , 2001, IEEE Trans. Inf. Theory.

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

[15]  Shlomo Shamai,et al.  Variations on the Gallager bounds, connections, and applications , 2002, IEEE Trans. Inf. Theory.

[16]  R. L. Dobrushin,et al.  Asymptotic Estimates of the Probability of Error for Transmission of Messages over a Discrete Memoryless Communication Channel with a Symmetric Transition Probability Matrix , 1962 .

[17]  N. R. Chaganty,et al.  Multidimensional strong large deviation theorems , 1996 .

[18]  Emre Telatar,et al.  Multi-access communications with decision feedback decoding , 1992 .

[19]  Aaron B. Wagner,et al.  The third-order term in the normal approximation for singular channels , 2013, 2014 IEEE International Symposium on Information Theory.

[20]  K. V. PETROVSKII,et al.  Limit theorems for large deviations of sums of independent lattice random vectors , 1996 .

[21]  Amiel Feinstein,et al.  A new basic theorem of information theory , 1954, Trans. IRE Prof. Group Inf. Theory.

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

[23]  Albert Guillén i Fàbregas,et al.  A derivation of the asymptotic random-coding prefactor , 2013, 2013 51st Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[24]  Aaron B. Wagner,et al.  Refinement of the Sphere-Packing Bound: Asymmetric Channels , 2012, IEEE Transactions on Information Theory.

[25]  Vincent Yan Fu Tan,et al.  A Tight Upper Bound for the Third-Order Asymptotics for Most Discrete Memoryless Channels , 2012, IEEE Transactions on Information Theory.

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

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

[28]  E.R. Berlekamp,et al.  The technology of error-correcting codes , 1980, Proceedings of the IEEE.

[29]  Albert Guillén i Fàbregas,et al.  Mismatched Decoding: Error Exponents, Second-Order Rates and Saddlepoint Approximations , 2013, IEEE Transactions on Information Theory.

[30]  Gregory Poltyrev,et al.  Bounds on the decoding error probability of binary linear codes via their spectra , 1994, IEEE Trans. Inf. Theory.