Abstract
We consider the implications of the equivalence of commutative semifields of odd order and planar Dembowski-Ostrom polynomials. This equivalence was outlined recently by Coulter and Henderson. In particular, following a more general statement concerning semifields we identify a form of planar Dembowski-Ostrom polynomial which must define a commutative semifield with the nuclei specified. Since any strong isotopy class of commutative semifields must contain at least one example of a commutative semifield described by such a planar polynomial, to classify commutative semifields it is enough to classify planar Dembowski-Ostrom polynomials of this form and determine when they describe non-isotopic commutative semifields. We prove several results along these lines. We end by introducing a new commutative semifield of order 38 with left nucleus of order 3 and middle nucleus of order 32.
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Coulter, R.S., Henderson, M. & Kosick, P. Planar polynomials for commutative semifields with specified nuclei. Des. Codes Cryptogr. 44, 275–286 (2007). https://doi.org/10.1007/s10623-007-9097-y
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DOI: https://doi.org/10.1007/s10623-007-9097-y