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Two-user DM-MAC code with perfect causal output feedback

Definition

For the finite DM-MAC of Two-user discrete memoryless multiple-access channel, fix integers $n \geq 1$ and $M_{1}, M_{2} \geq 1$ , and write $[m] = {1, \dots, m}$ . A perfect-feedback length- $n$ code has independent uniform messages $M_{j} \in [M_{j}]$ , deterministic encoders $f_{j i} : [M_{j}] \times Y^{i - 1} \to X_{j}$ for $j \in {1, 2}$ and $1 \leq i \leq n$ , and a joint decoder from $Y^{n}$ to the message pair. Both users see the same noiseless past output $Y^{i - 1} = (Y_{1}, \dots, Y_{i - 1})$ before use $i$ , with $Y^{0}$ the empty tuple; neither sees the other user's message. Rates are $R_{j} = n^{- 1} \ln M_{j}$ nats/use and error is average joint message error. Achievability means a sequence with $n \to \infty$ , vanishing error and limiting lower rates at least the specified pair; capacity is the closure.

Facts & Assumptions

Given: A finite DM-MAC, an integer $n \geq 1$ and two finite nonempty private message sets of sizes $M_{1}, M_{2} \geq 1$ .

[F1]

Two-user discrete memoryless multiple-access channel: its block transition law is the product of the single-use kernel $W (y ∣ x_{1}, x_{2})$ .

Proof

1.1givenF1

Given $(M_{1}, M_{2}, Y^{i - 1})$ , both causal maps determine $(X_{1 i}, X_{2 i})$ , and [F1] supplies a probability vector for $Y_{i}$ ; recursion over $i$ therefore gives a normalized joint law.

2.1givenstep 1.1algebra∎

Since $n \geq 1$ and $M_{j} \geq 1$ , each rate $n^{- 1} \ln M_{j}$ is a finite nonnegative real number. The joint-error event belongs to the finite probability space in step 1.1, so its probability is well-defined; the causal maps add no other random input to the model.

Depends on

Two-user discrete memoryless multiple-access channel

Used by

Cover-Leung inner bound for the two-user DM-MAC with feedback Theorem

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Abbas El Gamal and Young-Han Kim, Network Information Theory, Lecture Notes 18, Multiple Access Channel with Feedback (standard reference, not scraped)
Thomas M. Cover and Joy A. Thomas, Elements of Information Theory, 2nd ed., Chapter 15, §15.3 (standard reference, not scraped)