DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

No-feedback independent-private-message DM-MAC code

Definition

For a DM-MAC as in 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 length- $n$ no-feedback code consists of independent uniform private messages $M_{j} \in [M_{j}]$ , separate deterministic message-only encoders $f_{j} : [M_{j}] \to X_{j}^{n}$ , and one joint decoder $g : Y^{n} \to [M_{1}] \times [M_{2}]$ . The average joint error is $\Pr [g (Y^{n}) \neq (M_{1}, M_{2})]$ ; $R_{j} = n^{- 1} \ln M_{j}$ nats/use. A pair is achievable if a sequence with $n \to \infty$ has error tending to zero and limiting lower rates at least the pair; the capacity region is the closure. In particular, independent messages and separate encoders give $X_{1 i} ⊥ X_{2 i}$ at every coordinate $1 \leq i \leq n$ .

Facts & Assumptions

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

[F1]

Two-user discrete memoryless multiple-access channel: the conditional block law is $\prod_{i} W (y_{i} ∣ x_{1 i}, x_{2 i})$ .

Proof

1.1givenF1algebra

The independent uniform messages and the two deterministic maps induce separate input words; combining these with [F1] defines a probability law on the finite set of message, input and output tuples.

2.1givenstep 1.1algebra∎

Since each input coordinate is a function only of its own independent message, $X_{1 i}$ and $X_{2 i}$ are independent for $1 \leq i \leq n$ . Because $n \geq 1$ and $M_{j} \geq 1$ , each rate $n^{- 1} \ln M_{j}$ is a finite nonnegative real number. The decoder event belongs to the finite probability space in step 1.1, so its probability is well-defined.

Depends on

Two-user discrete memoryless multiple-access channel

Used by

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

Thomas M. Cover and Joy A. Thomas, Elements of Information Theory, 2nd ed., Chapter 15, §15.3 (standard reference, not scraped)
Abbas El Gamal and Young-Han Kim, Network Information Theory, Lecture Notes 4, Multiple Access Channels (standard reference, not scraped)