Correction to 'A Further Note on Backwards Markovian Models'

We wish to thank a conference reviewer and Prof. M. B. Pursley for pointing out an error in the arguments of the Appendix of the above note.' We forgot that two uncorrelated random variables may not remain so after conditioning. However, all the results in our note continue to be true, except that the arguments in the Appendix must be replaced by the more laborious, but correct, alternative proof outlined in the body of the paper. These proofs merely involve verifying that the appropriate orthogonality conditions are satisfied by the given expressions. More specifically, writing we have " xi +*,T=@xi+l+Gj3tlli+l,T-l for some 6?, 9, from which it is easy to show that so that (9) and (12) of the paper hold. It may similarly be shown that the II?;) are uncorrelated. This is the first comprehensive book on error-correcting codes written by mathematicians, and it establishes a standard of masterful scholarship that will not soon be surpassed. The authors have gleaned a treasury of coding facts (including an extensive table of best codes) from the 1478 entries in the bibliography and packaged them with clear explanations, elegant proofs, and graceful writing, together with many new or improved results. Despite its massiveness, this is a narrow book. It deals only with block codes, virtually only with independent errors and the Hamming metric, and principally with code structure as contrasted with methods for encoding and decoding. Have no doubt about it, this book presents error-correcting codes as seen by the eye of the combinatorial mathematician , not the communications engineer! It is a much closer relative to van Lint's slim monograph [ 1] than to the principal other books on algebraic coding [2]-[5]. Compared to these latter, this book probes much more deeply into such structural aspects of a code as its weight (or distance) distribution and its automorphism group (i.e., the group of permutations that take the code into itself). The authors are clearly excited by the rich variety of combinatorial configurations embodied in codes, and only an intransigent reader will avoid contagion. As no codes are richer in structure than the two Golay perfect codes and their extensions, these remarkable codes pop up every few pages in this book until the penultimate Chapter 20 (pp. 634-650), which is entirely devoted to showing their essential uniqueness and deriving their automorphism groups. Perhaps next in structural richness are maximum distance separable codes, …