Probability of error in information transmission over continuous-time white Gaussian channel with feedback

We consider a continuous-time white Gaussian channel with full feedback. The use of feedback link to improve the perfomance of communication systems has been studied by a number of authors. It is well known that the channel capacity is not increased by feedback, however error probability in information transmission can be substantially reduced by using feedback. The first remarkable work was done by Shalkwijk and Kailath (1966) where they proposed a feedback coding scheme and showed that the error probability goes down double exponentially fast. It was pointed out by Pinsker (1968) that one can construct a coding scheme for which the error probability decreases more rapidly than the exponential of any order. In this paper we propose a new feedback encoding scheme and a decoding scheme, and show that, under our coding sheme, the error probability goes down more rapidly than the exponential of any order.