Gelfand-Pinsker coding achieves the interference-free capacity

For a discrete memoryless channel with non-causal state information available only at the encoder, it is well-known that Gelfand-Pinsker coding achieves its capacity. In this paper, we analyze Gelfand-Pinsker coding scheme and capacity to bring out further understandings. We show that Gelfand-Pinsker capacity is equal to the interference-free capacity. Thus the capacity of a channel with non-causal state information available only at the encoder is the same as if the state information is also available at the decoder. Furthermore, the capacity-achieving conditional input distributions in these two cases are the same. This lets us connect the studied channel with state to the multiple access channel (MAC) with correlated sources and show that under certain conditions, the receiver can decode both the message and the state information. This dual decoding can be obtained in particular if the state sequences come from a known codebook with rate satisfying a simple constraint. In such a case, we can modify Gelfand-Pinsker coding by pre-building multiple codebooks of input sequences $X^n$, each codebook is for a given state sequence $S^n$, upon generating the auxiliary $U^n$ sequences. The modified Gelfand-Pinsker coding scheme achieves the capacity of the MAC with degraded message set and still allows for decoding of just the message at any state information rate. We then revisit dirty-paper coding for the Gaussian channel to verify our analysis and modified coding scheme.