Bit-Wise Sequential Coding With Feedback

This paper considers the problem of bit-wise channel coding over a Binary Symmetric Channel (BSC) with feedback. While it is known that feedback does not increase the capacity of a memoryless channel, it is believed to simplify the coding schemes for some channels. The most significant one is the Binary Erasure Channel (BEC) where capacity is achieved by a simple sequential bit-wise repetition code under which each bit is (re-)transmitted until it is received. This sequential bit-wise feedback code has the added advantage that it can be used for streaming applications over BECs. In contrast, there is no known sequential bit-wise code with feedback that can achieve non-zero transmission rate for a BSC. For example, under Posterior Matching for a binary input channel with feedback, also known as Horstein scheme, each message is considered in its entirety in a block coding manner.This paper proposes a sequential feedback coding scheme with a nested bit-wise structure that generalizes repetition codes. This scheme is shown to achieve strictly positive rate for a large class of binary input channels including a BSC with arbitrary cross-over probability p is an element of (0, 1/2). The analysis relies on characterizing a lower bound on the step-wise Extrinsic Jensen Shannon divergence.