Broadcast channel — superposition achievability (inner bound) #

The superposition (two-tier cloud / satellite) random-coding inner bound of a two-receiver broadcast channel, and its specialization to the achievability half of the degraded broadcast-channel coding theorem (Cover–Thomas Elements of Information Theory). The net-new tier relative to the multiple-access achievability (InformationTheory.Shannon.MAC) is the conditional (superposition) random codebook: satellite codewords are drawn from a conditional product law Πᵢ K(Uᵢ) steered by the cloud codeword, rather than a flat product law.

Main definitions #

• bcJointDistribution pU K W — the per-coordinate joint law of (U, X, Y₁, Y₂), U ∼ pU, X ∣ U ∼ K, (Y₁, Y₂) ∣ X ∼ W.
• bcInfo₂ pU K W — the cloud information I(U; Y₂) = H(U) + H(Y₂) − H(U, Y₂).
• bcInfo₁ pU K W — the satellite conditional information I(X; Y₁ ∣ U) = H(U, X) + H(U, Y₁) − H(U, X, Y₁) − H(U).
• BCCloudCodebook / BCSatelliteCodebook and their random-coding measures bcCloudCodebookMeasure, bcSatelliteCodebookMeasure, bcCodebookMeasure — the two-tier ensemble; the satellite law is a conditional product Πᵢ K(Uᵢ).

Main statements #

• bc_conditional_slice_prob_le — the gateway covering bound: the conditional-product mass of the jointly-typical satellite slice is ≤ exp(−n (I(X; Y₁ ∣ U) − ε)).
• bc_achievability_of_rate_lt — the class-free form: over any broadcast channel, a rate pair strictly inside the three superposition constraints is achievable.
• bc_achievability — the headline: any rate pair strictly inside the degraded-BC region R₁ < I(X; Y₁ ∣ U), R₂ < I(U; Y₂) is achievable with vanishing per-receiver error.

Module structure #

Umbrella of the Shannon/BroadcastChannel/Achievability/ family, re-exporting:

• Achievability.Setup — the ambient measure, the auxiliary-variable informations, the two-tier codebook, the positivity / marginal / relabeling infrastructure, the covering-bound ingredients, and the two-tier decoders.
• Achievability.ErrorAnalysis — the receiver-2 (cloud) and receiver-1 (strong) error analyses with their random-codebook averaging.
• Achievability.Assembly — the superposition random-coding assembly, the class-free bc_achievability_of_rate_lt, the rate-sum form bc_achievability_of_infoJoint_ge derived from it, and the degraded headline bc_achievability.