3-Designs from all Z4-Goethals-like codes with block size 7 and 8

Jyrki Lahtonen, Kalle Ranto*, Roope Vehkalahti

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

1 Citation (Scopus)

Abstract

We construct a family of simple 3 - (2m, 8, 14 (2m - 8) / 3) designs, with odd m ≥ 5, from all Z4-Goethals-like codes Gk. In addition, these designs imply the existence of other design families with the same parameters as the designs constructed from the Z4-Goethals code G1, i.e. the designs with a block size 7 by Shin, Kumar, and Helleseth and the designs with a block size 8 by Ranto. In the existence proofs we count the number of solutions to certain systems of equations over finite fields and use properties of Dickson and linearized polynomials. Also, the nonequivalence of the designs from different Goethals-like codes is considered.

Original languageEnglish
Pages (from-to)815-827
Number of pages13
JournalFinite Fields and Their Applications
Volume13
Issue number4
DOIs
Publication statusPublished - 1 Nov 2007
MoE publication typeA1 Journal article-refereed

Keywords

  • Dickson polynomial
  • Goethals code
  • Linearized polynomial
  • Quaternary code
  • t-Design

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