Commutative semifields and symplectic spreads.

*(English)*Zbl 1041.51002Recently, 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.

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.

Reviewer: Gary L. Ebert (Newark/Delaware)

##### MSC:

51A40 | Translation planes and spreads in linear incidence geometry |

51A35 | Non-Desarguesian affine and projective planes |

Full Text:
DOI

##### References:

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