Some families of zero- error block codes for the two-user binary adder channel with feedback

Abstract

Families of zero-error codes for the real binary adder channel with feedback that achieve high rate pairs are introduced. Two families of zero-error block codes are given for the case in which only one of the two senders receives feedback about the channel output. In the first of these families, the uninformed sender transmits at a rate of nearly one bit per symbol and the informed sender transmits slightly less that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</tex> bit per symbol. The second family is designed for the case in which the informed sender sends at or near one bit per symbol and the uninformed one sends nearly <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</tex> bit per symbol. A family of zero-error codes is introduced, based on the Fibonacci recursion; these codes are readily implemented by means of a simple square-dividing strategy. The Fibonacci codes achieve <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_{1}=R_{2}=\log_{2} [(1 + \sqrt{5})/2]</tex> in the limit of large block length. Time-sharing between members of these three code families is used to obtain an achievable rate region, or inner bound, to the zero-error capacity region for block coding. For the case in which the feedback is available to both senders, a variant of the Fibonacci difference equation is used to generate zero-error block codes with slightly higher asymptotic rate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R_{1}=R_{2}=0.717</tex> .