Reality bites

A.B. Calderbank, G. David Forney, Alexander Vardy

A 16-state tail-biting trellis for the binary (24,12,8) Golay code C/sub 24/ 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 C/sub 24/. Unwrapping this tail-biting trellis produces a periodically-time-varying 16-state rate- 1/2 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 1/2 . 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.

Masato Tajima, Koji Okino, Tatsuto Murayama

In this paper, we discuss the reduction of error-trellises for tail-biting convolutional codes. In the case where some column of a parity-check matrix has a monomial factor (with indeterminate D), we show that the associated tail-biting error-trellis can be reduced by cyclically shifting the corresponding error-subsequence by l (the power of D) time units. We see that the resulting reduced error-trellis is again tail-biting. Moreover, we show that reduction is also possible using backward-shifted error-subsequences.

Liesbeth C. Venema

Zhiliang Zhu, Songtao Liang

The well-known Forney algorithm, Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm and product construction for minimal linear conventional trellises have all successfully extended to tail-biting trellises. As first author of this paper, one of my paper, in a way, popularized Massey construction of minimal linear conventional trellises to tail-biting trellises, just referred as TB-Massey trellises. In this paper, we make some necessary and important optimization and improvement of TB-Massey trellises, mainly via elaborately constricting vertexes of the trellis graph as far as possible. Then display performance of all kinds of the extended tail-biting trellises by comparison. Though having not found the thorough generalized tail-biting Massey trellises, we get some positive progress and results