Qualified difference sets from unions of cyclotomic classes

Qualified difference sets (QDS) composed of unions of cyclotomic classes are discussed. An exhaustive computer search for such QDS and modified QDS that also possess the zero residue has been conducted for all powers n=4,6,8 and 10. Two new families were discovered in the case n=8 and some new isolated systems were discovered for n=6 and n=10

Lüke and power residue sequence diffusers

Conventional Schroeder diffusers have been successfully used for many years. However, their frequency range is limited by the flat plate effect that occurs when all the wells radiate in phase. This occurs at harmonics of p times the design frequency f(0), where p is the small prime that is used to generate the structure. A typical diffuser, using p=7 and f(0)=500 Hz, has an upper frequency limit of only 3.5 kHz. Achieving a first flat plate frequency above 20 kHz requires a prime equal to at least 41 and results in diffusers that are too big to be practical in most applications. This paper suggests an alternative approach using number theoretic sequences that, although short in length, are based on large integers. Two new sequences, Type-II Luke and power residue, have this desired characteristic. They are investigated using both simple models and the more exact boundary element method. The results show the flat plate effect is moved to much higher frequencies as expected. For Luke sequences at certain frequencies, redirection rather than dispersion is achieved. Modulation techniques can be used to mitigate these problems. Power residue sequences perform the best, providing good diffusion and a flat plate frequency outside the audible range