Tightened Upper Bounds on the ML Decoding Error Probability of Binary Linear Block Codes

Abstract

The performance of maximum-likelihood (ML) decoded binary linear block codes is addressed via the derivation of tightened upper bounds on their decoding error probability. The upper bounds on the block and bit error probabilities are valid for any memoryless, binary-input and outputsymmetric communication channel, and their effectiveness is exemplified for various ensembles of turbo-like codes over the AWGN channel. An expurgation of the distance spectrum of binary linear block codes further tightens the resulting upper bounds.

New Techniques for Upper-Bounding the ML Decoding Performance of Binary Linear Codes

“…As shown in [25,Appendix D], the average weight spectra of a random linear code [n, k] can be found as…”

“…One objective of this paper is, without too much complexity increase, to reduce the number of involved terms in the conventional union bound. The other objective of this paper is to tighten the bound on Pr{E d }, which used to be upper-bounded by the pair-wise error DRAFT October 25,2018 probability, where intersections of half-spaces related to codewords other than the transmitted one are counted more than once. For some of well-known existing improved bounds based on GFBT, such as the sphere bound (SB), the tangential-sphere bound (TSB) and the Divsalar bound, see the monograph [2, Ch.…”

“…We have interpreted the "truncated" union bound as an upper-bounding technique based on the GFBT, where the region R is defined by a sub-optimal decoding algorithm. To bound DRAFT October 25,2018 Pr{E, y ∈ R}, we have enlarged R to R n , as shown in the derivation from (14) to (15). The objective of this section is to reduce the effect of such an enlargement.…”

“…Though the ML decoding algorithm is prohibitively complex for most practical codes, tight bounds can be used to predict their performance without resorting to computer simulations. As shown in [1] [2], most bounding techniques have connections to either the 1965 Gallager bound [3][4][5][6] or the 1961 Gallager-Fano bound [7][8][9][10][11][12][13][14][15][16][17][18]. This paper is relevant to the 1961 Gallager-Fano bound, which is also called Gallager's first bounding technique (GFBT) in the literature.…”

Non-random coding error exponent for lattices

“…Many tight ML upper bounds originate from a general bounding technique, developed by Gallager [12]. Gallager's technique has been utilized extensively in literature [5], [13], [14]. Similar forms of the general bound, displayed hereafter, have been previously presented in literature [6], [15].…”