On Witsenhausen´s counterexample: The asymptotic vector case

Motivated by the presence of an implicit communication channel in the asymptotic version of Witsenhausen´s counterexample, implicit discrete memoryless channels (IDMC) with discrete memoryless (DM) states are considered. Information-theoretic lower and upper bounds (based respectively on the ideas from rate-distortion theory and hybrid-coding) are derived on the optimal distortion in estimating the input of the implicit channel. The intuition gained from the DMIC with DM state model is then used to evaluate the optimal distortion for the asymptotic version of the Witsenhausen counterexample. The minimum distortion is characterized for the counterexample and it is shown that a combination of linear coding and dirty-paper coding (DPC) proposed in [1] achieves the minimum distortion.