New Ring-Linear Codes from Geometric Dualization

In the 1960s and 1970s the Nordstrom-Robinson-Code [30] and subsequently the infinite series of the Preparata[31], Kerdock[21], Delsarte-Goethals[6] and Goethals-Codes [7] were discovered. Apart from a few corner cases, all of these codes are non-linear binary block codes that have higher minimum distance than any known comparable (having equal size and length) linear binary code. We will call such codes better-than-known-linear or BTKL. In [26, Research Problem (15.4)] the question was raised if the Preparata and Kerdock codes are better than any comparable linear binary code (better-than-linear, BTL1). For the Preparata series this was shown in [2]. Of the remaining above mentioned series still only the smaller codes are proven to be BTL. Later the striking discovery was made [28,12] that all these codes can be constructed as images of certain Z4-linear codes under the Gray map. Since then, a lot of research has been done on Z4-linear codes and on linear codes over more general finite rings. However, the examples on BTLor BTKL-codes found since then are comparatively sparse. They include Gray images of QR-codes over Z4, the two Calderbank-McGuire-codes [3,4] and some quasi-cyclic codes over Z4 [1]2. Two further examples [13,22] come from a hyperoval in the projective Hjelmslev plane over the 16-element Galois ring GR(16, 4) where a similar Gray map allows the comparison with F4-linear codes.