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Periodic Golay pairs and pairwise balanced designs. (English) Zbl 1493.05055

In this paper, a classification of periodic Golay pairs of length less than \(40\) is provided, making use of a relation between certain pairwise balanced designs with \(v\) points and periodic Golay pairs of length \(v\).
A pair \((a,b)\) of \(\{\pm 1\}\) sequences is a periodic Golay pair (PGP) if \[\sum_{i=0}^{v-1}a_i a_{i+s} + \sum_{i=0}^{v-1}b_i b_{i+s}=0\]
for every shift \(s\), with \(0\leq s\leq v-1\).
Periodic Golay pairs generalize Golay pairs, which are known to have applications in multi-slit spectroscopy, signal processing, digital communications, and other areas. PGPs exist at lengths where no Golay pairs may exist, and they also possess the properties required for the applications.
In this paper, the authors construct pairwise balanced designs (PBDs) using orbit matrices, with suitable parameters and given cyclic automorphism group, and they use them to construct PGPs. They classify PBDs with these conditions which correspond to PGPs of length \(v\) for all \(v \leq 34\), and they observe that there are no periodic Golay pairs of length \(36\) and \(38\). Moreover, to complete the classification of PBDs and PGPs up to isomorphism, a suitable notion of isomorphism is introduced. Finally, the authors show how the PGPs constructed and their related orbit matrices can be used to construct quasi-cyclic self-orthogonal linear codes over suitable finite fields.

MSC:

05B30 Other designs, configurations
05E18 Group actions on combinatorial structures
11B83 Special sequences and polynomials
94B05 Linear codes (general theory)

Software:

Magma; GAP
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References:

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