Lower bounds on reliability functions of variable-length nonsystematic convolutional codes for channels with noiseless feedback

Abstract

A variable-length, nonsystematic, convolutional encoding, and successive-decoding scheme is devised to establish significant improvements in the reliability functions of memoryless channels with noiseless decision feedback. It is shown that, for any but pathological discrete memoryless channels with noiseless feedback, there exists a variable-length convolutional code such that the reliability function of the channel can be bounded below by the channel capacity <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</tex> for all transmission rates less than <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</tex> . By employing a modified version of this scheme, it is also constructively shown that, for an additive-white-Gaussian-noise (AWGN) channel with noiseless feedback it is possible to find a variable-length convolutional code such that the channel reliability function can be bounded below by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\alpha_0 c_{infty}</tex> for all rates less than the channel capacity <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C_{\infty}</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\alpha_0 = max (1, \gamma/2)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\gamma</tex> is the maximum allowable expected-peak-to-expected-average-power ratio at the transmitter.