On the Number of Information Symbols in Difference-Set Cyclic Codes

The concept of a clifference-set cyclic code has been described by E. J. Weldon, Jr. in the preceding paper. 1 In Ref. 1 it is shown that such a code is almost as powerful as a Bose-Chaudhuri code and considerably simpler to implement. It is the purpose of this paper to determine some of the more important properties of this code and its dual code (cf. Sec. IV). It may be pointed out that the problems we consider are equivalent to determining certain properties of incidence matrices of Desarguesian planes. I I . S I M P L E D I F F E R E N C E SETS AND ASSOCIATED CYCLIC CODES A simple difference set S is a collection of I integers {rfi, · · · ,dt modulo n such that every a ^ 0 (mod n) can be uniquely expressed in the form rf, -- dj = a (mod n), for some d,, dj in S. Of course, n = 1(1 - 1) + 1. If d(x) (the difference-set polynomial) is defined by e(x) = E 1057 , 1062 T H E B E L L SYSTEM T E C H N I C A L J O U R N A L , S E P T E M B E R 19(36 then it follows that 71-1 d(x)d(x~1) = I + J2 i=i This may be written Q(x)0(x~l) =