Cyclic Codes over Finite Rings

A new Gray map between codes over finite rings R = F
₂
+ uF
₂
+ u
²
F
₂
and codes over F
₂
is defined. We prove that the Gray image of a linear (1+u+u
²
)-cyclic code over R of length n is a binary distance invariant linear cyclic code. We also prove that, if n is odd, then every binary code which is the Gray image of a linear cyclic code over R of length n is equivalent to a cyclic code.