New characterisations of the Nordstrom-Robinson codes

Understand

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.