Kerdock codes and related planes

Kantor, W.M., Kerdock codes and related planes, Discrete Mathematics 106/107 (1992) 297-302. Among the many aspects of coding theory Jack van Lint has studied intensively are some generalizations of Preparata and Kerdock codes (see Baker et al. (1983), Cameron and Van Lint (1991) and Van Lint (1983)). There are still many open problems concerning these. This note is a brief discussion of problems and new results involving orthogonal spreads, translation planes and associated generalized Kerdock codes. 1. Orthogonal spreads Let V be a vector space of dimension 4m over a finite field L of characteristic 2, where m 2 2. Assume that V is equipped with a quadratic form Q of Witt index 2m; the associated bilinear form is denoted (u, u). Then the pair V, Q is equivalent to the pair L4m, Qdm, where Write the standard ordered basis of L4m as e,, . . . , ezm,fi, . . . ,hrn, so that Qdm(ei) = Q4,Jfi) = (ei, ej) = (5, fi) = 0 and (e,, fi) = 6, for 1 c i, i s 2m. We will be concerned with totally singular 2m-spaces. Examples of these are E = (e,, . . .,ezm) andF=(f,,.. . , hm). Each totally singular 2m-space having only 0 in common with E looks like with M a skew-symmetric 2m x 2m matrix (1.1) Correspondence to: W.M. Kantor, Department of Mathematics, University of Oregon, Eugene, OR 97403, USA. * Supported in part by NSF and NSA grants. 0012-365X/92/$05.OOQ 1992-Elsevier Science Publishers B.V. All rights reserved