Almost difference sets have interesting applications in cryptography and coding theory. We give a well-rounded treatment of known families of almost difference sets, establish relations between some difference sets and some almost difference sets, and determine the numerical multiplier group of some families of almost difference sets. We also construct six new classes of almost difference sets, and four classes of binary sequences of period n/spl equiv/0 (mod 4) with optimal autocorrelation. We have also obtained two classes of relative difference sets and four classes of divisible difference sets (DDSs). We also point out that a result due to Jungnickel (1982) can be used to construct almost difference sets and sequences of period 4l with optimal autocorrelation.

In this correspondence we give several new families of binary sequences of period N with optimal three-level autocorrelation, where N/spl equiv/2 (mod 4). These sequences are either balanced or almost balanced. Our construction is based on cyclotomy.

We show that the cross correlation between two ternary m-sequences of period 3/sup n/-1 that differ by the decimation d=2/spl middot/3/sup m/+1, where n=2m+1, takes on three different values. We conjecture the same result for the decimation d=2/spl middot/3/sup r/+1, where n is odd and r is defined by the condition 4r+1/spl equiv/0 mod n. These two new cases form in a sense ternary counterparts of two previously confirmed binary cases, the conjectures of Welch and Niho (1972).

The existence of a ternary [28, 6, 16] code is considered. We show that without loss of generality, a generator matrix of such a code must satisfy certain conditions. A computer search over all matrices that satisfy these conditions reveals that a ternary [28, 6, 16] code does not exist.