Linear tail-biting trellises, the square-root bound, and applications for Reed-Muller codes

via time slices which involve only nine information symbols, viz. u t ; u t01 ; 1 1 1 ; u t08. From the time slices for the phases that are multiples of 4 it follows that information symbol u t08 does not contribute to the corresponding output. This information symbol does not contribute to the outputs in the following time instants either; hence, we do not have to store it! Thus the Kötter–Vardy convolutional encoder can be realized by only eight memory elements [11], although not in controller canonical form. (When we use time-varying convo-lutional codes to prove ensemble properties we consider realizations in controller canonical form with feedforward shift registers and time-varying connections [3]. For example, the Golay convolutional code can be encoded by a rate R = 1=2 time-varying encoder with four delay elements in controller canonical form.) We showed that the Kötter–Vardy convolutional code has d free = 12 and its first spectral components are phase n12 n16 n20 n24 Apart from the Golay convolutional code, the Kötter–Vardy convo-lutional code is the only Type II, rate R = 1=2, convolutional code known to us. Calderbank, Forney, and Vardy proved that a Type II, binary , time-invariant convolutional code of rate R = 1=2 does not exist ([1, Lemma 4]). The best rate R = 1=2, time-invariant convolutional code of memory m = 8 has [12] which is better than T KV (W) for high signal-to-noise ratios but worse for low signal-to-noise ratios, since n 13 = n 14 = n 15 = 0 in TKV (W). ACKNOWLEDGMENT The authors wish to thank G. D. Forney. Not only did he suggest the problem and act as a clearing house for the " Type II News Group, " but also, perhaps most importantly, he provided constant encouragement during their search for a needle in a (huge) haystack. They are also grateful to N. J. A. Sloane for identifying B2 and to R. Kötter for an illuminating discussion on the Kötter–Vardy convolutional code. Minimal tail-biting trellises: The Golay code and more, " IEEE Trans. The binary self-dual codes of length up to 32: A revised enumeration, " J.volutional encoders with an optimum distance profile, " IEEE Trans. In-Abstract—Linear tail-biting trellises for block codes are considered. By introducing the notions of subtrellis, merging interval, and sub-tail-biting trellis, some structural properties of linear tail-biting trellises are proved. It is shown that …