Distance Distribution of Binary Codes and the Error Probability of Decoding

Abstract

Abstract-We address the problem of bounding below the probability of error under maximum-likelihood decoding of a binary code with a known distance distribution used on a binarysymmetric channel (BSC). An improved upper bound is given for the maximum attainable exponent of this probability (the reliability function of the channel). In particular, we prove that the "random coding exponent" is the true value of the channel reliability for codes rate in some interval immediately below the critical rate of the channel. An analogous result is obtained for the Gaussian channel.

Sharpening of an Upper Bound for the Reliability Function of a Binary Symmetric Channel

“…To some extent, such combining was performed in [12,13]; however, the approach of [11] was used there not in the best way. Moreover, the most important proofs (concerning some variation problems) in [12,13] were based on purely numerical computations (without an analytical support), which makes it difficult to verify them.…”

“…Moreover, the most important proofs (concerning some variation problems) in [12,13] were based on purely numerical computations (without an analytical support), which makes it difficult to verify them.…”

On the joint source-channel coding error exponent for discrete memoryless systems

“…Moreover, closed-form parametric expressions of the improved lower bound and its corresponding condition are provided in [57] for systems with equidistant DMCs. 5 In light of the recent work in [11], where the random coding exponent E (R; W ) of the BSC is shown to be indeed the true value of the channel error exponent E(R; W ) for code rates R in some interval directly below the channel critical rate (in other words, it is shown that for the BSC with its " above a certain threshold, E (R; W ) = E(R; W ) for R R C where R can be less than R (W ) [11]), we note via (1) and the lower bound in (22) and (23) that region B B B where E is exactly known can be enlarged. B and C C C. In A A A, E = 0.…”

Upper bound on list-decoding radius of binary codes

“…Namely, we apply result of Kalai and Linial [11,Proposition 3.2]: For every β with h(β) ≤ R there exists a sequence ǫ n → 0 such that for every code C of rate R there is a ξ 0 satisfying (14) such that…”