There is exactly one Z2Z4-cyclic 1-perfect code

TL;DR

This paper proves the uniqueness of a specific Z2Z4-cyclic 1-perfect code of length 15 and shows that no other such codes exist, providing a parity check matrix and discussing extended codes.

Contribution

It establishes the non-existence of Z2Z4-cyclic 1-perfect codes except for one specific case and characterizes that code with a parity check matrix.

Findings

Only one Z2Z4-cyclic 1-perfect code exists at length 15.

Extended 1-perfect codes cannot be Z2Z4-cyclic.

Provides a parity check matrix for the unique cyclic code.

Abstract

Let ${\cal C}$ be a ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive code of length $n > 3$. We prove that if the binary Gray image of ${\cal C}$, $C=\Phi({\cal C})$, is a 1-perfect nonlinear code, then ${\cal C}$ cannot be a ${\mathbb{Z}}_2{\mathbb{Z}}_4$-cyclic code except for one case of length $n=15$. Moreover, we give a parity check matrix for this cyclic code. Adding an even parity check coordinate to a ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive 1-perfect code gives an extended 1-perfect code. We also prove that any such code cannot be ${\mathbb{Z}}_2{\mathbb{Z}}_4$-cyclic.