5.1 Introduction
This chapter extends VLSF codes introduced in Chapter 4 to DM-MACs.
Some prior work on VLF codes over MACs are as follows. Truong and Tan [72] extend the results in [18] to the Gaussian MAC under an average power constraint. Trillingsgaard et al. [73] study the VLSF scenario where a com- mon message is transmitted across a K-user discrete memoryless broadcast channel. Heidari et al. [74] extend Burnashev’s work to the DM-MAC and derive lower and upper bounds on the error exponents of VLF codes for the DM-MAC. Bounds on the performance of VLSF codes for the DM-MAC with an unbounded number of decoding times appear in [75]. The K-transmitter MAC achievability bounds from [72] and [75] employ 2K − 1 simultaneous information density threshold rules.
5.2 Definitions for MACs
We begin by introducing the definitions used for the multi-transmitter setting.
A K-transmitter DM-MAC is defined by a triple QK
k=1Xk, PYK|X[K],YK , where Xk is the finite input alphabet for transmitter k ∈ [K], YK is the finite output alphabet of the channel, and PYK|X[K] is the channel transition probability.
Let PYK denote the marginal output distribution induced by the input dis- tribution PX[K]. The unconditional and conditional information densities are defined for each non-empty A ⊆[K] as
ıK(xA;y)≜lnPYK|XA(y|xA)
PYK(y) (5.1)
ıK(xA;y|xAc)≜lnPYK|X[K](y|x[K])
PYK|XAc(y|xAc), (5.2) where Ac = [K]\ A. Note that in (5.1)–(5.2), the information density func- tions depend on the transmitter setA unless further symmetry conditions are assumed (e.g., in some cases we assume that the components ofPX[K] are i.i.d., and PYK|X[K] is invariant to permutations of the inputs X[K]).
The corresponding mutual informations under the input distributionPX[K] and the channel transition probability PYK|X[K] are defined as
IK(XA;YK)≜E[ıK(XA;YK)] (5.3) IK(XA;YK|XAc)≜E[ıK(XA;YK|XAc)]. (5.4) The dispersions are defined as
VK(XA;YK)≜Var [ıK(XA;YK)] (5.5) VK(XA;YK|XAc)≜Var [ıK(XA;YK|XAc)]. (5.6) For brevity, we define
IK ≜Ik(X[K];YK) (5.7)
VK ≜Var
ıK(X[K];YK)
. (5.8)
A VLSF code for the MAC with K transmitters is defined similarly to the VLSF code for the PPC.
Definition 5.2.1. Fix ϵ ∈ (0,1), N ∈ (0,∞), and positive integers Mk, k ∈ [K]. An (N, L, M[K], ϵ) VLSF code for the MAC comprises
1. integer-valued decoding 0≤n1 <· · ·< nL,
2. K finite alphabets Uk, k ∈ [K], defining common randomness random variables U1, . . . , UK,
3. K sequences of encoding functions fn(k):Uk×[Mk]→ Xk, k∈[K], 4. a stopping time τ ∈ {n1, . . . , nL} for the filtration generated by
{U1, . . . , UK, YKnℓ}Lℓ=1, satisfying an average decoding time constraint (4.5), and
5. Ldecoding functionsgnℓ:U[K]×YKnℓ →
K
Q
k=1
[Mk]∪{e}forℓ∈[L], satisfying an average error probability constraint
P
gτ(U[K], YKτ)̸=W[K]
≤ϵ, (5.9)
where the independent messages W1, . . . , WK are uniformly distributed on the sets [M1], . . . ,[MK], respectively.
5.3 Achievability Bounds
Our main results are second-order achievability bounds for the rates approach- ing a point on the sum-rate boundary of the MAC achievable region increased by a factor of 1−ϵ1 . Theorem 5.3.1, below, is an achievability bound for the asymptotic regime L=O(1).
Theorem 5.3.1. Fix ϵ ∈ (0,1), an integer L=O(1)≥2, and distributions PXk, k ∈ [K], and arbitrary constants a(A) ∈ (0, IK(XA;YK|XAc)) for A ∈ P([K]). For any K-transmitter DM-MAC (QK
k=1Xk, PYK|X[K],YK), there ex- ists an (N, L, M[K], ϵ) VLSF code with
X
k∈[K]
lnMk≤ N IK 1−ϵ −
r
Nln(L−1)(N) VK 1−ϵ +O
s N ln(L−1)(N)
!
, (5.10)
X
k∈A
lnMk≤ N(IK(XA;YK|XAc)−a(A))
1−ϵ +o(N) (5.11)
for all A ∈ P([K]).
Theorem5.3.1 follows from an application of the non-asymptotic achievability bound, Theorem 5.3.2, below.
Theorem 5.3.2. Fix constantsϵ∈(0,1),γ,λ(A)>0forA ∈ P([K]), integers 0 ≤ n1 < · · · < nL, and distributions PXk, k ∈ [K]. For any DM-MAC with K transmitters (QK
k=1Xk, PYK|X[K],YK), there exists an (N, L, M[K], ϵ) VLSF code with
ϵ≤P h
ıK(X[K]nL;YKnL)< γi
(5.12) +
K
Y
k=1
(Mk−1) exp{−γ} (5.13)
+
L
X
ℓ=1
X
A∈P([K])
P
ıK(XAnℓ;YKnℓ)> N(IK(XA;Y) +λ(A))
(5.14)
+ X
A∈P([K])
Y
k∈Ac
(Mk−1)
!
exp{−γ+N IK(XA;YK) +N λ(A)} (5.15)
N ≤n1+
L−1
X
ℓ=1
(nℓ+1−nℓ)P h
ıK(X[K]nℓ;YKnℓ)< γ i
. (5.16)
Proof sketch: The proof of Theorem5.3.2uses a random coding argument that employs K i.i.d. codebook ensembles with distributions PXk, k ∈ [K]. The receiver employsLdecoders that operate by comparing an information density ıK(x[K]n;yn) for each possible transmitted codeword set to a threshold. At time nℓ, decoder gnℓ computes the information densities ıK(X[K]nℓ(m[K]);YKnℓ); if there exists a message vector mˆ[K satisfying ıK(X[K]nℓ( ˆm[K]);YKnℓ) > γ, then the receiver decodes to the message vectormˆ[K]. Otherwise, the decoder emits output e, and the receiver passes the decoding time nℓ without decoding.
If nℓ < nL, the transmission continues until the next decoding time. The term (5.12) bounds the probability that the information density corresponding to the true messages is below the threshold for all decoding times; (5.13) bounds the probability that all messages are decoded incorrectly; and (5.14)- (5.15) bound the probability that the messages from the transmitter index set A ⊆[K]are decoded incorrectly, and the messages from the index set Ac are decoded correctly.
In the application of Theorem 5.3.2 to prove Theorem 5.3.1, we choose the parameters λ(A) and γ so that the terms in (5.14)-(5.15) decay exponentially with N, which become negligible compared to (5.12) and (5.13). Between (5.12) and (5.13), the term (5.12) is dominant whenL does not grow withN, and (5.13) is dominant when L grows linearly with N.
Like the single-threshold rule from [85] for the RAC, the single-threshold rule employed in the proof of Theorem 5.3.2 differs from the decoding rules em- ployed in [72] for VLSF codes over the Gaussian MAC with expected power constraints and in [75] for the DM-MAC. In both [72] and [75], L = ∞, and the decoder employs 2K −1 simultaneous threshold rules for each of the boundaries that define the achievable region of the MAC withK transmitters.
Those rules fix thresholds γ(A), A ∈ P([K]), and decode messages m[K] if for allA ∈ P([K]), the codeword for m[K] satisfies
ıK(XAnℓ(mA);YKnℓ|XAnℓc(mAc))> γ(A), (5.17) for some γ(A), A ∈ P([K]). Our decoder can be viewed as a special case of (5.17) obtained by setting γ(A) =−∞ for A ̸= [K].
Analyzing Theorem 5.3.2 in the asymptotic regime L = Ω(N), we determine
that there exists an (N,∞, M[K], ϵ)VLSF code if (5.11) holds and X
k∈[K]
lnMk ≤ N IK
1−ϵ −lnN +O(1). (5.18) Here, the asymptotic regime between Land N (e.g.,L = Ω(N)or L=O(1)) is important rather than whether L < ∞ or L = ∞. This is because if we truncate an infinite-length code at time n = 2N, by Chernoff bound, the resulting penalty term added to the error probability decays exponentially with N, whose effect in (5.18) is o(1). Therefore, for any VLSF code, L=∞ case can be treated as L = Ω(N) regardless of the number of transmitters.
See Appendix D.2.1for the proof of (5.18).
ForL=∞, Trillingsgaardet al. [75] numerically evaluate their non-asymptotic achievability bound for a DM-MAC while Truong and Tan [72] provide an achievability bound with second-order term −O(√
N)for the Gaussian MAC with average power constraints. Applying our single-threshold rule and analy- sis to the Gaussian MAC with average power constraints improves the second- order term in [72] from−O(√
N)to−lnN+O(1) for all non-corner points in the achievable region. The main challenge in [72] is to derive a tight bound on the expected value of the maximum over A ⊆ [K] of stopping times τ(A) for the corresponding threshold rules in (5.17). In our analysis, we avoid that chal- lenge by employing a single-threshold decoder whose average decoding time is bounded by E
τ([K]) .
Under the same model and assumptions onL, to achieve non-corner rate points that do not lie on the sum-rate boundary, which corresponds to a(A) = 0 in (5.11) for one or more A ∈ P([K]), we modify our single-threshold rule to (5.17), where A is the transmitter index set corresponding to the capacity region’s active sum-rate bound at the (non-corner) point of interest. Following steps similar to the proof of (5.18) gives second-order term −lnN +O(1) for those points as well. For corner points, more than one boundary is active1; therefore, more than one threshold rule in (5.17) is needed at the decoder. In this case, again for L=∞, [72] proves an achievability bound with a second- order term−O(√
N). Whether this bound can be improved to−lnN+O(1) as in (5.18) remains an open problem.
1The capacity region of a K-transmitter MAC is characterized by the region bounded by2K−1 planes. By definition of a corner point, at least two inequalities corresponding to these planes are active at a corner point.
C h a p t e r 6