The generator matrix

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

generates a code of length 69 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+6x^68+100x^69+6x^70+1x^72+12x^73+1x^74+1x^82

The gray image is a code over GF(2) with n=552, k=7 and d=272.
This code was found by Heurico 1.16 in 0.297 seconds.