Nonhomogeneous Trellis codes for the Quasi-Synchronous Multiple-Access Binary adder channel with Two Users

Abstract

A trellis code is {\em homogeneous} if the number of branches emanating from each node (or state) in the trellis diagram is constant. For example, convolutional codes are linear homogeneous trellis codes. A trellis code is {\em nonhomogeneous} if the number of branches emanating from each node in the trellis diagram is not the same. The two-user binary adder channel is a multiple-access channel with two binary inputs, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x_{1}</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x_{2}</tex> , and one ternary output, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y = x_{1} + x_{2}</tex> , where the addition is done in the real number field. The adder channel is synchronous if both encoders and the decoder maintain block (frame) synchronism. It is quasi-synchronous if the encoders do not start their blocks at the same time, but the decoder knows the position of each block. The difference between the starting times of the blocks is called the slippage. The channel is asynchronous if no block synchronism exists among the encoders and the decoder. Some uniquely decodable code pairs <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(C_{1}, C_{2})</tex> are presented that can be used to transmit information reliably over the quasi-synchronous binary adder channel with two users. One of the codes is a nonhomogeneous trellis code, the other is a common block code. Our code rates are better than Deaett-Wolf codes and are close to or equal to the asymptotic rates of Kasami {\em et al}. A method for calculating the rates of nonhomogeneous trellis codes is described. An algorithm for finding more uniquely decodable code pairs for the quasi-synchronous binary adder channel is formulated.