Ivan Mogilnykh: Weak isometries of Preparata codes

The talk is about some structural properties of one important class of error-correcting codes, namely Preparata codes.

Let C₁ and C₂ be codes with code distance d. A mapping J:C₁ → C₂, such that for any x, y from C₁ the equality d(x, y) = d holds if and only if d(J(x), J(y)) = d is called a weak isometry. Obviously two codes are weakly isometric if and only if the minimal distance graphs of these codes are isomorphic.

We prove that any weak isometry of two (punctured) Preparata codes is an isometry. As a consequence, two Preparata codes of length n, n ≥ 2¹², have isomorphic minimum distance graphs if and only if these codes are equivalent. The analogous result is obtained for punctured Preparata codes of length at least 2¹⁰ − 1.