Commutative Semifields of Order 3⁵

Abstract

A concrete construction of a commutative semifield of order 3⁵ is introduced. This semifield is proved to be not isotopic to 𝔽_3⁵ or Albert's twisted fields of such an order. It is shown that it is equivalent up to the action of the symmetric group S ₃ neither to any of those semifields nor to Coulter–Matthews or Ding–Yuan semifields with 3⁵ elements. Also, a complete description of commutative semifields of order 3⁵ is provided, with the help of computational tools. As a consequence, the existence of a completely new semifield is revealed.

Key Words:

• Commutative
• Computational methods
• Finite semifield
• Planar function

2000 Mathematics Subject Classification:

• Primary 17D99
• Secondary 17-04, 17-08

ACKNOWLEDGMENTS

I. F. Rúa was partially supported by MTM2010-18370-C04-01 and 1B08-147. E. F. Combarro was supported by MEC-TIN-2007-61273 and MICINN-TTN-2010-14971.

Notes

Communicated by A. Elduque.