Asymptotic Upper Bounds on the Minimum Distance of Trellis Codes

Abstract

A trellis code is a "sliding window" method of encoding a binary data stream as a sequence of signal points in R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> . When a trellis code is used to encode data at the rate of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> bits/channel symbol, each channel input depends not only on the most recent block of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> bits to enter the encoder, but will also depend on a set of ν bits preceding this block. The ν bits determine the state of the encoder and the most recent block of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> bits generates the channel symbol conditional on the encoder state. The performance of a trellis code depends on a suitably defined minimum distance property of that code. This paper obtains upper bounds on this minimum distance that are better than any previously known.