Abstract
We construct a family of simple 3-(2m,8,14(2m–8)/3) designs, with odd m≥5, from all Z 4-Goethals-like codes \({\mathcal{G}}_k\) with k=2l and l≥1. In addition, these designs imply also the existence of the other design families constructed from the Z 4-Goethals codes \({\mathcal{G}}_1\) by Ranto. In the existence proofs we count the number of solutions to certain systems of equations over finite fields and use Dickson polynomials and variants of cyclotomic polynomials and identities connecting them.
Part of the results have been published in the dissertation of the second author [13].
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Lahtonen, J., Ranto, K., Vehkalahti, R. (2006). 3-Designs from Z 4-Goethals-Like Codes and Variants of Cyclotomic Polynomials. In: Ytrehus, Ø. (eds) Coding and Cryptography. WCC 2005. Lecture Notes in Computer Science, vol 3969. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11779360_6
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DOI: https://doi.org/10.1007/11779360_6
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