Some new optimum Golomb rulers

0 Theorem 4, unlike Theorem 2, is not a s'pecial case of a known general result: it is not known if the statement of Theorem 4 holds for twoaccess channels that are not pairwise reversible. There is a converse to Theorem 4: For any fixed rate strictly inside R o , there exists a code with that rate for which X I 1 + o(N). This result holds for general twoaccess channels, and is proved by random-coding [9], [lo]. Thus for PR twoaccess channels , R,, is the critical region for A. (Whether the same holds in general remains unsettled.) We have defined R,, as the union of an uncountable number of regions. Unfortunately no simpler characterization of R, (such as the single-letter characterization that exists in the case of ordinary channels) has been found. The difficulty here is that for twoaccess channels no analog of Lemma 2 exists. For more on open problems in this area, see [lo] and [8]. code has rate 2 (R I , R r)). Iv. APPLICATIONS TO SEQUENTIAL DECODING Consider sequential decoding of a tree code on a one-user channel. Assume that the tree code is infinite in length and that each path in the tree is equally likely to be the true (transmitted) path. Let C, denote the expected number of computational steps for the sequential decoder to decode correctly the first N branches of the tree code. We take the asymptotic value of C,/N as a measure of complexity for sequential decoding. We say that a rate R is achievable by sequential decoding if there exists a tree code with rate R for which C , / N remains bounded as N goes to infinity. The supremum of achievable rates is called the cutoff rate and denoted by RLonlp. The link between the complexity of sequential decoding and lower bounds to X is established by the following idea of [4]. Lemma 4: Consider a sequence of block codes obtained by truncating a given tree code at level N, N 21. Let A, denote the average list size for the Nth code in this sequence. Then I CN/N 2 XN/2. (22) This lemma and Theorem 2 imply that, for sequential decoding on ordinary PR channels at rates R > R,, C,/N goes to infinity with increasing N. This implies in turn that for such channels For all one-user channels (pairwise reversible or not), …