Abstract
It is well known that associated with a translation plane π there is a family of equivalent spreads. In this paper, we prove that if one of these spreads is symplectic and π is finite, then all the associated spreads are symplectic. Also, using the geometric intepretation of the Knuth’s cubical array, we prove that a symplectic semifield spread of dimension n over its left nucleus is associated via a Knuth operation to a commutative semifield of dimension n over its middle nucleus.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
André J (1954). Über nicht-desarguessche ebenen mit transitive translationgruppe. Math Z 60: 156–186
Ball S, Brown M (2004). The six semifield planes associated with a semifield flocks. Adv Math 189: 68–87
Ball S, Bamberg J, Lavrauw M, Penttila T. (2004). Symplectic spreads. Des Codes Cryptogr 32: 9–14
Dembowski P (1968). Finite Geometries. Springer, Berlin Heidelberg New York
Hughes DR, Piper FC (1972) Projective planes. Springer
Kantor WM (1982). Spreads, translation planes and kerdock sets, I. Siam J Algebra Discr Methods 3: 151–165
Kantor WM (2003). Commutative semifields and symplectic spreads. J Algebra 270: 96–114
Kantor WM, Williams ME (2004). Symplectic semifields planes and \(\mathbb{Z}_4\)-linear codes Trans Am Soc 356: 895–938
Knuth DE (1965). Finite semifields and projective planes. J Algebra 2: 182–217
Lunardon G (1999). Normal spreads. Geom Dedicata 75: 245–261
Lüneburg H (1980). Translation planes. Springer, Berlin Heidelberg New York
Mellinger KE (2003). A geometric relationship between equivalent spreads. Des Codes Cryptogr 30: 63–71
Maschietti A (2003). Symplectic translation planes and line ovals. Adv Geom 3: 123–143
Segre B (1964). Teoria di Galois, fibrazioni proiettive e geometrie non desarguesiane. Ann Mat Pur Appl 64: 1–76
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Lunardon, G. Simplectic spreads and finite semifields. Des. Codes Cryptogr. 44, 39–48 (2007). https://doi.org/10.1007/s10623-007-9054-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-007-9054-9