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Symplectic semifield planes and Z4-linear codes (2004)
| Venue: | TRANSACTIONS OF THE AMERICAN MATHENATICAL SOCIETY |
| Citations: | 15 - 6 self |
BibTeX
@ARTICLE{Kantor04symplecticsemifield,
author = {William M. Kantor and Michael E. Williams},
title = {Symplectic semifield planes and Z4-linear codes},
journal = {TRANSACTIONS OF THE AMERICAN MATHENATICAL SOCIETY},
year = {2004},
volume = {356},
number = {3},
pages = {895--938}
}
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Abstract
There are lovely connections between certain characteristic 2 semifields and their associated translation planes and orthogonal spreads on the one hand, and Z4–linear Kerdock and Preparata codes on the other. These inter– relationships lead to the construction of large numbers of objects of each type. In the geometric context we construct and study large numbers of nonisomorphic affine planes coordinatized by semifields; or, equivalently, large numbers of non–isotopic semifields: their numbers are not bounded above by any polynomial in the order of the plane. In the coding theory context we construct and study large numbers of Z4–linear Kerdock and Preparata codes. All of these are obtained using large numbers of orthogonal spreads of orthogonal spaces of maximal Witt index over finite fields of characteristic 2. We also obtain large numbers of “boring ” affine planes in the sense that the full collineation group fixes the line at infinity pointwise, as well as large numbers of Kerdock codes “boring ” in the sense that each has as small an automorphism group as possible. The connection with affine planes is a crucial tool used to prove inequivalence theorems concerning the orthogonal spreads and associated codes, and also to determine their full automorphism groups.
Keyphrases
large number z4-linear code orthogonal spread symplectic semifield plane affine plane preparata code maximal witt index full automorphism group coding theory context non isotopic semifields lovely connection infinity pointwise orthogonal space z4 linear kerdock nonisomorphic affine plane geometric context automorphism group translation plane crucial tool full collineation group kerdock code finite field inequivalence theorem inter relationship
