Symplectic semifield planes and ${\mathbb Z}_4$–linear codes
William M. Kantor, Michael E. Williams
2003
Transactions of the American Mathematical Society
There are lovely connections between certain characteristic 2 semifields and their associated translation planes and orthogonal spreads on the one hand, and Z 4 -linear Kerdock and Preparata codes on the other. These interrelationships 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
more »
... re not bounded above by any polynomial in the order of the plane. In the coding theory context we construct and study large numbers of Z 4 -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. producing new connections with other areas of mathematics. The present paper focuses further on the finite geometry aspects of these codes: once we have obtained suitable planes and orthogonal spreads, the machinery developed in [CCKS] has immediate coding-theoretic consequences. We briefly introduce some of the terminology used throughout this paper. Binary Kerdock codes are constructed using Kerdock sets: families of 2 n−1 skew-symmetric n×n binary matrices such that the difference of any two is nonsingular. Orthogonal spreads (in our setting this means families of q m + 1 totally singular (m + 1)-spaces of an orthogonal space of type O + (2m + 2, q) that partition the singular points of the space) arise from analogous sets of (m + 1) × (m + 1) matrices over GF(q) for any q. Symplectic spreads (families of q m + 1 totally isotropic m-spaces of a 2m-dimensional symplectic space over GF(q) that partition the points of the space) arise in a similar way from symmetric matrices, and produce both affine planes and Z 4 -Kerdock codes. Various aspects of the similarities of the descriptions of these combinatorial objects were thoroughly investigated in [CCKS]; we refer to that paper and [Ka3] for further background. For now we only mention the Gray map, an isometry φ from Z N 4 to Z 2N 2 that was used so effectively in [HKCSS] for passing between binary and Z 4 -codes. We will construct binary Kerdock codes K 2 for which K 4 = φ −1 (K 2 ) is Z 4 -linear. The corresponding Z 4 -Preparata code P 4 = K ⊥ 4
doi:10.1090/s0002-9947-03-03401-9
fatcat:rtl6oa4qvfbbjdftxjxw6pndyi