Broadcast channel — the degraded converse at the ambient of a code #

Physical degradedness of a broadcast channel says every letter of receiver 2's output is a stochastic function of the corresponding letter of receiver 1's output. On the ambient measure of a block code this upgrades to a statement about whole blocks: receiver 2's output block is appended to receiver 1's by the blockwise product of the degrading kernel, so at every letter the receiver-2 prefix is conditionally independent of the current receiver-1 letter given message 2 and the receiver-1 prefix. That conditional independence is exactly the structural hypothesis of the degraded converse, which therefore lands at a bare broadcast code.

Main statements #

• bcConverse_degradedBlock — under physical degradedness the letter-i output of receiver 1 is conditionally independent of the receiver-2 prefix given message 2 and the receiver-1 prefix, on the ambient measure of a broadcast code. This is the hypothesis h_deg_block of bc_degraded_converse at bcConverseAmbient.
• bc_degraded_converse_from_code — the degraded outer bound (Cover–Thomas) at a bare broadcast code, with the auxiliary Uᵢ = (W₂, Y₂^{i-1}) read off the ambient.

Implementation notes #

Degradedness enters only through the degrading kernel it provides, and that kernel is applied to the whole output block before the block identity is cut down to a prefix: bcConverse_block_append appends the entire receiver-2 block to (messages, Y₁ⁿ) by the blockwise product piBlockKernel of that kernel, and bcConverse_prefix_append reindexes the resulting identity along the injection of Fin i into Fin n. The block form is the one the product structure of the ambient gives directly, so keeping the two steps apart separates the product-measure recombination from the reindexing.

Appending the degraded output block #

The degraded converse at a code #