The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  1  X  X  X  1  X  X  X  1  1  1  1  1  X  X  X 2X+2 2X+2 2X+2  X 2X+2  1 2X+2 2X+2 2X+2  X  X  X  X  1  1  X  1  1 2X+2  1
 0 2X  0  0  0 2X 2X 2X  0  0  0  0 2X 2X 2X 2X  0  0  0 2X  0 2X 2X 2X  0 2X 2X  0  0 2X 2X  0  0  0 2X 2X  0  0 2X 2X  0 2X 2X  0  0 2X 2X  0  0 2X  0  0 2X  0 2X
 0  0 2X  0 2X 2X 2X  0  0  0 2X 2X 2X 2X  0  0  0  0 2X 2X 2X  0  0 2X  0 2X 2X 2X 2X  0  0  0  0 2X 2X  0  0 2X  0 2X 2X 2X  0 2X  0 2X  0 2X  0 2X  0 2X 2X  0  0
 0  0  0 2X 2X  0 2X 2X  0 2X 2X  0  0 2X 2X  0  0 2X 2X  0  0 2X  0 2X 2X  0 2X 2X  0 2X  0  0 2X 2X  0  0 2X 2X  0  0  0 2X 2X  0  0 2X 2X  0 2X  0  0 2X 2X  0 2X

generates a code of length 55 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 54.

Homogenous weight enumerator: w(x)=1x^0+7x^54+106x^55+7x^56+1x^62+6x^63

The gray image is a code over GF(2) with n=440, k=7 and d=216.
This code was found by Heurico 1.16 in 0.078 seconds.