The Zero-Error Feedback Capacity of State-Dependent Channels

Abstract

The zero-error feedback capacity of the Gelfand-Pinsker channel is established. It can be positive even if the channel's zero-error capacity is zero in the absence of feedback. Moreover, the error-free transmission of a single bit may require more than one channel use. These phenomena do not occur when the state is revealed to the transmitter causally, a case that is solved here using Shannon strategies. Cost constraints on the channel inputs or channel states are also discussed, as is the scenario where-in addition to the message-also the state sequence must be recovered.

Zero-Error Feedback Capacity for Bounded Stabilization and Finite-State Additive Noise Channels

“…Here we show that this upper bound is, in fact, achievable. 1 Unlike [4], [5], we do not assume that channel state information is available at the encoder or decoder. Examples, including a Gilbert-Elliott channel, are considered for which the explicit value of C 0f is computed.…”

“…The uniform zero-error feedback code is more restrictive than well-known zero-error feedback code used in literature where the transmission always starts at time 1, i.e., t 0 = 0; see, e.g., [4], [5], [41]. However, this only implies that the encoding strategy does not change over time, and this condition is imposed to be able to use the zero-error feedback code starting from any time t 0 + 1 ∈ N. This gives the code a uniform (or time-invariance) property, and yet the code must be able to convey the messages without error for the corresponding block.…”

“…In [4], a dynamic programming formulation for computing C 0f was introduced for a finite-state channel modelled as a Markov decision process assuming that state information is available at both encoder and decoder. In [5], Gel'fand-Pinsker channels having i.i.d. internal states are studied, and single-letter formula for C 0f are derived assuming that the states are known at the transmitter.…”

Fundamental Limits of Communication Over State-Dependent Channels With Feedback

“…We refer the reader to [9] for a review of this area. The work most closely related to what we discuss here is that of Bracher-Lapidoth [2], where the zero-error feedback capacity of state-dependent channels was determined, under the assumptions that fixed-length encoding is being used and that state information is available only at the encoder. We also mention the works of Zhao-Permuter [18], where the authors gave a characterization of the zero-error feedback capacity under fixed-length encoding for channels with state information at both the encoder and the decoder (but in which the state process is not necessarily memoryless and is even allowed to depend on the channel inputs), and Tallini-Al-Bassam-Bose [16], where zero-error VLF communication over the binary Z-channel was studied.…”

“…We note that, unlike for DMCs[14], the conditions for positivity of the zero-error capacity under fixed-length coding for SD-DMCs are in general not the same in the presence or absence of feedback. E.g., there exist channels with C nc,-0,f,FL > C nc,-0,-,FL = 0; see[2, Thm 7]. Hence, the situation is more subtle when channels have states, and not all results are straightforward generalizations of their DMC counterparts.…”