Finding finite $B_2$-sequences faster

Abstract:

A $B_2$-sequence is a sequence $a_1<a_2<\cdots <a_r$ of positive integers such that the sums $a_i+a_j$, $1\le i\le j\le r$, are different. When $q$ is a power of a prime and $\theta$ is a primitive element in $GF(q^2)$ then there are $B_2$-sequences $A(q,\theta )$ of size $q$ with $a_q<q^2$, which were discovered by R. C. Bose and S. Chowla. In Theorem 2.1 I will give a faster alternative to the definition. In Theorem 2.2 I will prove that multiplying a sequence $A(q,\theta )$ by integers relatively prime to the modulus is equivalent to varying $\theta$. Theorem 3.1 is my main result. It contains a fast method to find primitive quadratic polynomials over $GF(p)$ when $p$ is an odd prime. For fields of characteristic 2 there is a similar, but different, criterion, which I will consider in “Primitive quadratics reflected in $B_2$-sequences”, to appear in Portugaliae Mathematica (1999).