Codes based on a trellis cut-set transformation. I. Rotationally invariant codes

Abstract

Let a trellis section /spl Tscr/ generate a trellis code /spl Cscr/. We study two trellis sections based on /spl Tscr/, a trellis section /spl Tscr//sub cs/ and a trellis section /spl Tscr//sub de/. We show that /spl Tscr/ can be transformed to a cut-set trellis section /spl Tscr//sub cs/, which is equivalent to /spl Tscr/ in the sense that both /spl Tscr/ and /spl Tscr//sub cs/ generate /spl Cscr/ and both /spl Tscr/ and /spl Tscr//sub cs/ have the same decoding complexity. A differential encoder trellis section is equivalent to the trellis section obtained by following /spl Tscr/ with a differential encoder. It is shown that both /spl Tscr//sub cs/ and /spl Tscr//sub de/ have inverse transform trellis sections. A differential encoder trellis section generates a rotationally invariant (RI) code in a particularly simple and straightforward way. But an RI code need not have a differential encoder trellis section. However, for all of the RI codes examined here, we show that the cut-set trellis section can be arranged into a differential encoder trellis section. This means that these codes can be decomposed into an encoder followed by a differential encoder. Further we show that when /spl Tscr/ is formed using a linear binary convolutional encoder and a mapping by set partitioning, then /spl Tscr/ followed by a differential encoder gives an RI code which in some cases is as good as the best previously known codes, after applying the inverse transform to /spl Tscr//sub de/.