The Art of Doing Science and Engineering: Learning to Learn

by Richard Hamming

Going out of time sequence, but still in idea sequence, I was once asked by at&t how to code things when humans were using an alphabet of 26 letters, ten decimal digits, plus a “space.” This is typical of inventory naming, parts naming, and many other namings of things, including the naming of buildings. I knew from telephone dialing error data, as well as long experience in hand computing, that humans have a strong tendency to interchange adjacent digits (a 67 is apt to become a 76), as well as change isolated ones (usually doubling the wrong digit, for example a 556 is likely to emerge as 566). Thus single error detecting is not enough. I got two very bright people into a conference room with me and posed the question. Suggestion after suggestion I rejected as not good enough until one of them, Ed Gilbert, suggested a weighted code. In particular he suggested assigning the numbers (values) 0, 1, 2, …, 36 to the symbols 0, 1, …, 9, A, B, …, Z, space. Next he computed not the sum of the values but if the kth symbol has the value (labeled for convenience) sk, then for a message of n symbols we compute “modulo” meaning divide this weighted sum by 37 and take only the remainder. To encode a message of n symbols, leave the first symbol, k = 1, blank, and whatever the remainder is, which is less than 37, subtract it from 37 and use the corresponding symbol as a check symbol, which is to be put in the first position. Thus the total message, with the check symbol in the first pos...