Association schemes related to universally optimal configurations, Kerdock codes and extremal Euclidean line-sets

H. Cohn et. al. proposed an association scheme of 64 points in R^{14} which is conjectured to be a universally optimal code. We show that this scheme has a generalization in terms of Kerdock codes, as well as in terms of maximal real mutually unbiased bases. These schemes also related to extremal line-sets in Euclidean spaces and Barnes-Wall lattices. D. de Caen and E. R. van Dam constructed two infinite series of formally dual 3-class association schemes. We explain this formal duality by constructing two dual abelian schemes related to quaternary linear Kerdock and Preparata codes.Comment: 16 page

Z₄-linear codes

Coding theory is a branch of mathematics that began in the 1940\u27s to correct errors caused by noise in communication channels. It is know that certain nonlinear codes satisfy the MacWilliams Identity. Much research has been done to explain this relation- ship. In the early 1990\u27s, five coding theorists discovered that nonlinear codes have linear properties if viewed under the alphabet Z4 rather than the usual alphabet F2. In 1994, these coding theorists published their results in a joint paper in IEEE Transactions on Information Theory. In this study, linear codes and nonlinear codes are introduced and characterized as Z4-linear codes to understand the importance of this discovery

New characterisations of the Nordstrom–Robinson codes

In his doctoral thesis, Snover proved that any binary $(m,256,\delta)$ code is equivalent to the Nordstrom-Robinson code or the punctured Nordstrom-Robinson code for $(m,\delta)=(16,6)$ or $(15,5)$ respectively. We prove that these codes are also characterised as \emph{completely regular} binary codes with $(m,\delta)=(16,6)$ or $(15,5)$, and moreover, that they are \emph{completely transitive}. Also, it is known that completely transitive codes are necessarily completely regular, but whether the converse holds has up to now been an open question. We answer this by proving that certain completely regular codes are not completely transitive, namely, the (Punctured) Preparata codes other than the (Punctured) Nordstrom-Robinson code