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

Let a trellis section 𝒯 generate a trellis code 𝒞. We study two trellis sections based on 𝒯, a “cut-set” trellis section 𝒯_cs and a “differential encoder” trellis section 𝒯_de. We show that 𝒯 can be transformed to a cut-set trellis section 𝒯_cs, which is equivalent to 𝒯 in the sense that both 𝒯 and 𝒯 _cs generate 𝒞 and both 𝒯 and 𝒯_cs have the same decoding complexity. A differential encoder trellis section is equivalent to the trellis section obtained by following 𝒯 with a differential encoder. It is shown that both 𝒯_cs and 𝒯_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 𝒯 is formed using a linear binary convolutional encoder and a mapping by set partitioning, then 𝒯 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 𝒯_de