Bounds for the capacity error function for unidirectional channels with noiseless feedback

Abstract

In digital systems such as fiber optical communications the ratio between probability of errors of type 1 → 0 and 0 → 1 can be large. Practically, one can assume that only one type of errors can occur. These errors are called asymmetric. Unidirectional errors differ from asymmetric type of errors, here both 1 → 0 and 0 → 1 type of errors are possible, but in any submitted codeword all the errors are of the same type.We consider q-ary unidirectional channels with feedback and give bounds for the capacity error function. It turns out that the bounds depend on the parity of the alphabet q. Furthermore we show that for feedback the capacity error function for the binary asymmetric channel is different from the symmetric channel. This is in contrast to the behavior of that function without feedback.

Coding with Noiseless Feedback over the Z-channel

“…In this paper, we develop new encoding algorithms for the Z-channel with feedback. In particular, we provide a family of error-correcting codes with the asymptotic rate (1 + τ )(1 − h(τ /(1 + τ ))), which is positive for any τ < 1 and improves the result from [16] in all but countable number of points. The corresponding lower bound on R(τ ) is shown in blue in Figure 2.…”

“…By random arguments, it was proved in [15] that R(τ ) > 0 for τ < 1/2. In [16] a feedback strategy based on the rubber method [3] was introduced to find an encoding strategy achieving a positive asymptotic rate for any τ < 1/2. The corresponding lower bound on R(τ ) is plotted in green on Figure 2.…”