code rate of 4/7), and this code is capable of correcting any error that occursin a single bit position of the codeword. This code was actually developed byHamming, Shannon’s co-worker at the time at Bell Labs. Because of patentconsiderations, however, Hamming did not describe and publish a paperabout his code until 1950.  The motivation for Hamming’s work was pri-marily to develop error-correcting and error-detecting codes for computersto increase reliability, not for point-to-point communication systems.Over the past 60 years or so, coding theory has become a highly devel-oped subject, and practical codes that can achieve reliable performance atrates approaching capacity on a number of important channels have beendeveloped. For many years, following Shannon’s seminal 1948 papers, error-correcting codes were studied mainly by mathematicians, and the subjecthad little commercial importance. In fact, the first IEEE CommunicationTheory Workshop held April of 1971 in St.Petersburg, Florida becameknown as the “coding is dead” workshop. Bob Lucky, who for many yearswrote a column for the IEEE Spectrum magazine, humorously describedthis meeting in one of his columns as follows: 45More precisely the noise power spectral density, i.e., the noise power per unit band-width. In general, the noise power spectral density will be a function of frequency. Thenoise is said to be ‘white’ when the noise power spectral density is a constant independentof frequency. Thermal noise, one of the most common sources of noise, is essentially white.
1.3. BRIEF HISTORICAL SKETCH45“A small group of us in the communications field will alwaysremember a workshop held in Florida about 20 years ago... Oneof my friends [Ned Weldon] gave a talk that has lived in infamyas the “coding is dead” talk.His thesis was that he and theother coding theorists formed a small, inbred group that hadbeen isolated from reality for too long. He illustrated this talkwith a single slide showing a pen of rats that psychologists hadpenned in a confined space for an extensive period of time.Icannot tell you what those rats were doing, but suffice it to saythat the slide has since been borrowed many times to depict thedepths of depravity into which such a disconnected group canfall... ”One of the first major uses of digital communication techniques, includ-ing error-correcting codes, was for deep space communications. In particu-lar, the 1968 Pioneer probe space mission. For this mission, the propagationdistances involved were enormous, resulting in high propagation losses andvery weak signal strengths at the receiver. In order to maintain reliable com-munications at a reasonable data rate between the probe and earth-boundstations, error-correcting codes were needed to obtain power-efficient oper-ation.A second major use of digital communication techniques, startingin 1986, was for data modems that connecting remote users to computersvia analog telephone lines. The deep-space channel has plenty of bandwidthbut very little signal power. The telephone channel, on the other hand, has