IEEE Information Theory Society Newsletter Vol. 51, No. 4, December 2001 Editor: Lance C. Pérez ISSN 1059-2362 2001 ISIT Plenary Lecture Are Turbo-like Codes Effective on Nonstandard Channels?* Robert J. McEliece California Institute of Technology Pasadena, California USA E-mail:
[email protected] Abstract sition at R = C. For R < C, the minimum attainable p is 0+, Turbo codes and their close relatives, e.g. Gallager and RA whereas for RC>=,/p H−1 1 −CR, where H (x)is codes, have been shown to be extraordinarily effective on the min 2 () 2 the binary entropy function. ”standard”channel models, e.g. the binary erasure, binary symmetric, and the AWGN channels. In this paper we discuss Shannon’s theorem is an existence theorem, and as engi- the possibility of using turbo-like codes on a variety of other neers we naturally ask, ”how hard is it to communicate channel models, including some that are nonsymmetric, at rate R and decoded error probability p?” We will re- nonbinary, and multiuser. Our basic conclusion is that binary strict ourselves to the case R < C, for if R > C we would turbo-like codes with graph-based iterative decoding have need to consider source coding, which is beyond the great potential on nearly all such channels. scope of this paper. We are especially interested in com- municating reliably at rates very near capacity, so let us 1. How Hard is it to Approach Channel assume in fact that Capacity? RC=−()1⑀, We begin at the beginning, with Shannon’s celebrated where ⑀ is a small positive number.