Abstract
The automorphism group of the Zetterberg code Z of length 17 (also a quadratic residue code) is a rank three group whose orbits on the coordinate pairs determine two strongly regular graphs equivalent to the Paley graph attached to the prime 17. As a consequence, codewords of a given weight of Z are the characteristic vectors of the blocks of a PBIBD with two associate classes of cyclic type. More generally, this construction of PBIBDs is extended to quadratic residue codes of length \(\equiv 1 \pmod {8},\) to the adjacency codes of triangular and lattice graphs, and to the adjacency codes of various rank three graphs. A remarkable fact is the existence of 2-designs held by the quadratic residue code of length 41 for code weights 9 and 10.
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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue: On Coding Theory and Combinatorics: In Memory of Vera Pless”
This research is supported by the National Natural Science Foundation of China (12071001), the Excellent Youth Foundation of Natural Science Foundation of Anhui Province (1808085J20), and the Research Council of Norway under Grant 247742/O70.
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Shi, M., Wang, S., Helleseth, T. et al. Quadratic residue codes, rank three groups and PBIBDs. Des. Codes Cryptogr. 90, 2599–2611 (2022). https://doi.org/10.1007/s10623-021-00918-z
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DOI: https://doi.org/10.1007/s10623-021-00918-z