Other Helper Capacities

The Listsize Capacity of the Gaussian Channel with Decoder Assistance

“…In this section we prove the direct part of Theorem 2: when the decoder can be provided with a rate- description of the noise, the convergence (19) can be achieved at all transmission rates below . As noted earlier, the converse follows directly from (Remark 4 in [ 3 ]).…”

“…The analogous result for the modulo-additive channel was proved in [ 3 ] by having the helper provide the decoder with a lossless description of the type of the noise sequence. Since this type fully specifies the a posteriori probability of the transmitted message, the decoder’s remotely-plausible-with-this-help list contains only messages whose a posteriori probability is equal to that of the correct message.…”

“…The “converse,” that (19) cannot be achieved when the rate exceeds the RHS of (20), follows from (Remark 4 in [ 3 ]). The “direct part,” describing a coding scheme that achieves (19) with rates approaching the RHS of (20), is proved in Section 5 .…”

“…The relation (1) is reminiscent of the analogous result on the erasures-only capacity of the Gaussian channel with a rate- helper (Remark 10 in [ 3 ]), namely, that where C denotes the Shannon capacity of the Gaussian channel (without help) (Theorem 9.1.1 in [ 4 ]), and is the erasures-only capacity, which is defined like but with the requirement on the -th moment of the list replaced by the requirement that the list be of size 1 with probability tending to one. (The Gaussian erasures-only capacity with a helper is given by the RHS of (4) irrespective of whether the assistance is provided to the encoder or decoder.)…”

“…In proving (1), we shall focus on the “direct part,” i.e., that the right-hand side (RHS) of (1) is achievable. The “converse,” that no rate exceeding the RHS of (1) is achievable, is omitted because it follows directly from (Remark 4 in [ 3 ]): There it is shown that this is true even if, given the received sequence and the provided help, the list contains only a subset of the messages that are of positive a posteriori probability, namely, those that are a posteriori at least as likely as the transmitted message.…”