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Commutative semifields and symplectic spreads. (English) Zbl 1041.51002
Recently, much attention is again being paid to semi-fields; that is, algebras satisfying all of the axions for a skew field except (possibly) associativity. One of the earliest papers on semi-fields was [D.E. Knuth, J. Algebra 2, 182-217 (1965; Zbl 0128.25604)], in which Donald Knuth defined objects called cubical arrays.
In the paper under review the author uses these cubical arrays to provide a bijection between commutative semi-field planes and symplectic semi-field planes. While the number of previously known commutative semi-field planes of any given order $$N$$ was less than log $$N$$, the number constructed in this remarkable paper is not bounded above by any polynomial in $$N$$, where here the order $$N$$ of the planes constructed is even. Thus, the gauntlet has been thrown down, and there are now many different (mutually non-isomorphic) commutative semi-field planes known in characteristic 2, but not nearly so many in odd characteristic. However, as the author points out, there are more types of constructions for commutative semi-fields in odd characteristic, and one still would like more examples in characteristic 2 having odd dimension over GF(2) because of the connection with coding theory.

##### MSC:
 51A40 Translation planes and spreads in linear incidence geometry 51A35 Non-Desarguesian affine and projective planes
##### Keywords:
commutative semi-field planes; symplectic spreads
Full Text:
##### References:
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