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Mathematics of Computation

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Finding finite $B_2$-sequences faster

Author: Bernt Lindström
Journal: Math. Comp. 67 (1998), 1173-1178
MSC (1991): Primary 11B75, 11Y55, 12E20
MathSciNet review: 1484901
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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).

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Additional Information

Bernt Lindström
Affiliation: Department of Mathematics, Royal Institute of Technology, S-100 44, Stockholm, Sweden
Address at time of publication: Turbingränd 18, S-17675 Järfälla, Sweden

Keywords: $B_2$-sequence, Bose-Chowla theorem, finite field, primitive element, primitive quadratic
Received by editor(s): November 21, 1996
Article copyright: © Copyright 1998 American Mathematical Society