Upper Bounds on the Minimum Distance of Trellis Codes

Abstract

A trellis code is a “sliding window” method of encoding a binary data stream into a sequence of real numbers that are input to a noisy transmission channel. When a trellis code is used to encode data at the rate of k bits/channel symbol, each channel input will depend not only on the most recent block of k data bits to enter the encoder but will also depend on, say, the v bits preceding this block. The v bits determine the state of the encoder and the most recent block of k bits generates the channel symbol conditional on the encoder state. The performance of trellis codes, like that of block codes, depends on a suitably defined minimum-distance property of the code. In this paper we obtain upper bounds on this minimum distance that are simple functions of k and v. These results also provide a lower bound on the number of states required to achieve a specific coding gain.