\(\mathbb{Z}_4\)-Kerdock codes, orthogonal spreads, and extremal Euclidean line-sets. (English) Zbl 0916.94014

Calderbank, A. R.; Cameron, P. J.; Kantor, W. M.; Seidel, J. J.

Proc. Lond. Math. Soc., III. Ser. 75, No. 2, 436-480 (1997).

It is known that some Kerdock and Preparata codes can be obtained in a very simple way as binary images under a certain map, called the Gray map, of linear codes over \(\mathbb{Z}_4\). One of the main results in the paper is an explanation of the Gray map in terms of suitable finite groups and geometries. The paper gives many nice and interesting connections between the geometry of binary orthogonal and symplectic vector spaces on the one hand and extremal real and complex line-sets having only two angles on the other.
Symplectic spreads in a binary vector space determine \(\mathbb{Z}_4\)-Kerdock codes, while orthogonal spreads determine binary Kerdock codes. It is shown that the geometric/group-theoretic map from symplectic spreads to orthogonal spreads induces the Gray map from a corresponding Kerdock code over \(\mathbb{Z}_4\) to its binary image.
Examples of Kerdock and Preparata codes are given. Large number of inequivalent \(\mathbb{Z}_4\)-Kerdock codes are constructed. Some new \(\mathbb{Z}_4\)-linear Preparata codes are also constructed.

Reviewer: T.Helleseth (Bergen)

MSC:

94B60	Other types of codes
51M15	Geometric constructions in real or complex geometry
94B05	Linear codes (general theory)

Keywords:

Kerdock codes; \(Z_4\)-linear codes; Preparata codes; Gray map; orthogonal spreads