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4-Kerdock codes, orthogonal spreads, and extremal euclidean line-sets

Published online by Cambridge University Press:  01 September 1997

AR Calderbank
Affiliation:
AT&T Bell Laboratories, Murray Hill, NJ 07974, USA
PJ Cameron
Affiliation:
School of Mathematical Sciences, Queen Mary and Westfield College, London E1 4NS, UK
WM Kantor
Affiliation:
Department of Mathematics, University of Oregon, Eugene, Oregon 97403, USA
JJ Seidel
Affiliation:
Faculty of Mathematics and Computing Science, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands
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Abstract

When $m$ is odd, spreads in an orthogonal vector space of type $\Omega^+ (2m+2,2)$ are related to binary Kerdock codes and extremal line-sets in $\RR^{2^{m+1}}$ with prescribed angles. Spreads in a $2m$-dimensional binary symplectic vector space are related to Kerdock codes over $\ZZ_4$ and extremal line-sets in $\CC^{2^m}$ with prescribed angles. These connections involve binary, real and complex geometry associated with extraspecial 2-groups. A geometric map from symplectic to orthogonal spreads is shown to induce the Gray map from a corresponding $\ZZ_4$-Kerdock code to its binary image. These geometric considerations lead to the construction, for any odd composite $m$, of large numbers of $\ZZ_4$-Kerdock codes. They also produce new $\ZZ_4$-linear Kerdock and Preparata codes.

1991 Mathematics Subject Classification: primary 94B60; secondary 51M15, 20C99.

Type
Research Article
Copyright
London Mathematical Society 1997

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