On Golay polynomial pairs (Q1189706)

A dual pair consists of two polynomials \(A\) and \(B\) solving the equation \(A(z)\cdot A(z^{-1})+B(z)\cdot B(z^{-1})=\text{const}\). If also the coefficients of \(A\) and \(B\) are \(\pm 1\) then the pair is called a Golay pair. Golay pairs are known to find applications in optics, Fourier analysis and combinatorics. The complete determination of all possible Golay pairs is still an open problem; it is even unknown for which degrees Golay pairs exist. On the other hand this classification is made easier by the existence of a multiplication for Golay pairs which turn the set of possible solutions into a non-commutative monoid; since this monoid is mapped homomorphically into the multiplicative monoid of the positive integers we know that the possible degrees form a multiplicatively closed subset. This set contains the numbers 2, 10 and 26. It is also known that prime factors of degrees are incongruent \(3 \bmod 4\). The authors contribute to the theory and a more complete classification by associating to a Golay pair two more dual pairs, called the penultimate and ante-penultimate pair, respectively. These pairs have coefficients \(\pm 1\) or 0. The number of 0's is called the gap-number. The results show that these zero's cannot be distributed arbitrarily within the polynomials of the (ante) penultimate pair. The resulting structure is investigated and used subsequently in order to determine the possible Golay pairs with small gap numbers and small degrees exhaustively. Part of the job is done by hand; some more involved cases have been solved with computer support. The results are listed in the extensive tables completing the paper.