Bounds on the error probability of maximum likelihood decoding of a binary linear code are considered. The bounds derived use the weight spectrum of the code and they are tighter than the conventional union bound in the case of large noise in the channel. The bounds derived are applied to a code with an average spectrum, and the result is compared to the random coding exponent. The author shows that the bound considered for the binary symmetrical channel case coincides asymptotically with the random coding bound. For the case of AWGN channel the author shows that Berlekamp's (1980) tangential bound can be improved, but even this improved bound does not coincide with the random coding bound, although it can be very close to it.>Bounds on the error probability of maximum likelihood decoding of a binary linear code are considered. The bounds derived use the weight spectrum of the code and they are tighter than the conventional union bound in the case of large noise in the channel. The bounds derived are applied to a code with an average spectrum, and the result is compared to the random coding exponent. The author shows that the bound considered for the binary symmetrical channel case coincides asymptotically with the random coding bound. For the case of AWGN channel the author shows that Berlekamp's (1980) tangential bound can be improved, but even this improved bound does not coincide with the random coding bound, although it can be very close to it.>

IEEE Transactions on Information TheoryJul 1, 1994230 citations

ON THE PROBABILITY OF UNDETECTED ERROR FOR ITERATED CODES

Osamu Nagata, Ra S Awas

Jan 1, 19830 citations

On the probability of undetected error of linear block codes

Tadao Kasami, Shu Lin, Torleiv Kløve

Jan 1, 19824 citations

On Approximation, Bounding & Exact Calculation of Block Error Probability for Random Code Ensembles

Ralf R. Müller

This paper presents a method to calculate the exact average block error probability of some random code ensembles under maximum-likelihood decoding. The proposed method is applicable to various channels and ensembles. The focus is on both spherical and Gaussian random codes on the additive white Gaussian noise channel as well as binary random codes on both the binary symmetric channel and the binary erasure channel. While for the uniform spherical ensemble Shannon, in 1959, argued with solid angles in $N$ -dimensional space, the presented approach projects the problem into two dimensions and applies standard trigonometry. This simplifies the derivation and also allows for the analysis of the independent identically distributed (i.i.d.) Gaussian ensemble which turns out to perform better for short blocklengths and high rates. Moreover, a new lower bound on the average block error probability of the uniform spherical ensemble is found. For codes with more than three codewords, it is tighter than the sphere packing bound, but requires exactly the same computing effort. Furthermore, tight approximations are proposed to simplify the computation of both the exact average error probability and the two bounds. For the binary symmetric channel and the binary erasure channel, bounds on the average block error probability for i.i.d.\ random coding are derived and compared to the exact calculations.This paper presents a method to calculate the exact average block error probability of some random code ensembles under maximum-likelihood decoding. The proposed method is applicable to various channels and ensembles. The focus is on both spherical and Gaussian random codes on the additive white Gaussian noise channel as well as binary random codes on both the binary symmetric channel and the binary erasure channel. While for the uniform spherical ensemble Shannon, in 1959, argued with solid angles in $N$-dimensional space, the presented approach projects the problem into two dimensions and applies standard trigonometry. This simplifies the derivation and also allows for the analysis of the independent identically distributed (i.i.d.) Gaussian ensemble which turns out to perform better for short blocklengths and high rates. Moreover, a new lower bound on the average block error probability of the uniform spherical ensemble is found. For codes with more than three codewords, it is tighter than the sphere packing bound, but requires exactly the same computing effort. Furthermore, tight approximations are proposed to simplify the computation of both the exact average error probability and the two bounds. For the binary symmetric channel and the binary erasure channel, bounds on the average block error probability for i.i.d.\ random coding are derived and compared to the exact calculations.

arXiv:2003.06807Mar 15, 20204 citations

On Approximation, Bounding & Exact Calculation of Block Error Probability for Random Codes

Ralf R. Müller

This paper presents a method to calculate the exact average block error probability of some random code ensembles under maximum-likelihood decoding. Deviating from Shannon's 1959 solid angle argument, we project the problem into two dimensions and apply standard trigonometry. This enables us to also analyze Gaussian random codes in additive white Gaussian noise and binary random codes for the binary symmetric channel. We find that the Voronoi regions harden doubly-exponential in the blocklength and utilize that to propose the new median bound that outperforms Shannon's 1959 sphere packing bound for the uniform spherical ensemble, whenever the code contains more than three codewords. Furthermore, we propose a very tight approximation to simplify computation of both exact error probability and the two bounds.This paper presents a method to calculate the exact average block error probability of some random code ensembles under maximum-likelihood decoding. Deviating from Shannon's 1959 solid angle argument, we project the problem into two dimensions and apply standard trigonometry. This enables us to also analyze Gaussian random codes in additive white Gaussian noise and binary random codes for the binary symmetric channel. We find that the Voronoi regions harden doubly-exponential in the blocklength and utilize that to propose the new median bound that outperforms Shannon's 1959 sphere packing bound for the uniform spherical ensemble, whenever the code contains more than three codewords. Furthermore, we propose a very tight approximation to simplify computation of both exact error probability and the two bounds.

GlobecomDec 1, 20202 citations

On the Probability of Incorrect Decoding for Linear Codes

Marco Frego

Lecture notes in computer scienceJan 1, 20173 citations

N

-dimensional space, the presented approach projects the problem into two dimensions and applies standard trigonometry. This simplifies the derivation and also allows for the analysis of the independent identically distributed (i.i.d.) Gaussian ensemble which turns out to perform better for short blocklengths and high rates. Moreover, a new lower bound on the average block error probability of the uniform spherical ensemble is found. For codes with more than three codewords, it is tighter than the sphere packing bound, but requires exactly the same computing effort. Furthermore, tight approximations are proposed to simplify the computation of both the exact average error probability and the two bounds. For the binary symmetric channel and the binary erasure channel, bounds on the average block error probability for i.i.d.\ random coding are derived and compared to the exact calculations.