The Random Coding Bound Is Tight for the Average Linear Code or Lattice

Abstract

In 1973, Gallager proved that the random-coding bound is exponentially tight for the random code ensemble at all rates, even below expurgation. This result explained that the random-coding exponent does not achieve the expurgation exponent due to the properties of the random ensemble, irrespective of the utilized bounding technique. It has been conjectured that this same behavior holds true for a random ensemble of linear codes. This conjecture is proved in this paper. Additionally, it is shown that this property extends to Poltyrev's random-coding exponent for a random ensemble of lattices. I. INTRODUCTIONThe error exponent, for a particular channel, is a function describing the exponential decay rate (with increasing block length) of the maximum-likelihood decoding error probability, for any communication rate R below the capacity C. The random-coding exponent is constructed [1] by upper bounding the average of a maximum-likelihood decoder's error probability over a random ensemble of codes, and considering its exponential decay rate. In general, the randon-coding exponent has two distinct regions, separated by the critical rate R cr :1) The straight line region: 0 < R < R cr 2) The sphere packing region: R cr ≤ R < C The random-coding exponent is tight in the second region. This is easily shown via its equality to the sphere packing exponent, which is the exponential decay rate of a lower bound on the error probability in that region [1]. In the first region, there exists an expurgation rate 0 < R ex < R cr , such that through expurgation of "bad" codewords, an exponent better than the random-coding exponent is achievable for any rate 0 < R < R ex [1]. Naturally, this gives rise to the question, "Why is the random-coding exponent not tight in the region 0 < R < R cr ? Is it due to the poor performance of the random ensemble at low rates, or perhaps is it due to the upper bounding technique used for its construction?". The question is answered in Gallager's 1973 paper [2], where a lower bound on the average error probability of the random ensemble is shown, whose exponential rate coincides with the random-coding exponent at all rates. Evidently, the random-coding exponent at low rates is not tight due to the poor performance of the random ensemble rather than a poor bounding technique.The random-coding exponent, shown for random codes, applies for random linear codes as well.

A Joint Typicality Approach to Compute–Forward

“…where p q = Unif(F q ). The general joint distribution of the codewords resulting from this construction can be found in [48,Theorem 1].…”

Arıkan meets Shannon: Polar codes with near-optimal convergence to channel capacity

“…The described communication is just a transmission of a random linear code C = {vG, v ∈ {0, 1} k } through W ℓ , where the rate of the code is R = k ℓ ≤ I(W ) − ℓ −1/2 log 3 ℓ, so it is separated from the capacity of the channel. It is a well-studied fact that random (linear) codes achieve capacity for BMS, and moreover a tight error exponent was described by Gallager in [Gal65] and analyzed further in [BF02], [For05], [DZF16]. Specifically, one can show P e ≤ exp(−ℓE r (R, W )), where P e is the probability of decoding error, averaged over the ensemble of all linear codes of rate R, and…”

Arikan meets Shannon: polar codes with near-optimal convergence to channel capacity

“…The main theorem (Theorem 1.1) follows directly from combining the results described in the previous sections. Namely, by using (6) in (4) with ≤ ℓ −1/2+5 from (7), we derive…”

“…where the rate of the code is = ℓ ≤ ( ) − ℓ −1/2 log 3 ℓ, so it is separated from the capacity of the channel. It is a wellstudied fact that random (linear) codes achieve capacity for BMS, and moreover a tight error exponent was described by Gallager in [12] and analyzed further in [3], [10], [6]. Specifically, one can show ≤ exp(−ℓ ( , )), where is the probability of decoding error, averaged over the ensemble of all linear codes of rate , and ( , ) is the so-called random coding exponent.…”