Minimal tail-biting trellises: the Golay code and more

A 16-state tail-biting trellis for the binary (24,12,8) Golay code C₂₄ is exhibited. This tail-biting trellis is minimal. It has many other nice properties, and may be simpler to decode than the minimal conventional trellis for C₂₄. Unwrapping this tail-biting trellis produces a periodically-time-varying 16-state rate-½ convolutional code. This “Golay convolutional code” is the first known example of a self-dual doubly-even binary linear convolutional code. It has greater minimum distance than any 16-state time-invariant convolutional code of rate ½. Demonstrably minimal tail-biting trellises are given for several other binary and nonbinary self-dual codes. For some nonbinary codes, tail-biting trellises based on code generators over a group are simpler than any tail-biting trellis based on linear generators over a field