Generalized Reed-Muller Codes

the possible choices for n and k are rather thinly distributed in the class of all pairs (n, k) with k ~ n--and it is, therefore, often inefficient to make use of such codes in concrete situations (that is, when a desired pair (n, k) is far from any achievable pair). We have succeeded in overcoming this difficulty by generalizing the Reed-Muller codes in such a way that they exist for every pair (n, k). I t should perhaps be emphasized that for given n and k our procedure enables us (with very little effort) to exhibit (n, k) codes explicitly in terms of a basis and to know their error-correcting capabilities. We shall illustrate this assertion in the course of the discussion by constructing a (26, 11) code which turns out to be 3-error-correcting. As a rule, the distance properties of our codes are not as good as those of the Bose-Chaudhuri codes (see Bose and Ray-Chaudhuri (1960), Gorenstein and Zierler (1961), Peterson (1961), and Weiss (1960)). This seems to be due to the fact that our codes depend on a simple kind of additive averaging and that multiplieative properties do not enter. The decoding procedure rests on the customary majoriW testing (Reed, 1953). We also discuss the canonical form, that is, where the code vectors consist of k information bits followed by n k parity check bits.