The generator matrix

 1  0  0  1  1  1 2X  1  1  0  1  1  2 X+2  1 3X+2 3X  X  1  1  1  1  1 2X  1  1 3X  1  X  1 X+2  1  1  1  1 3X+2  2  1 3X+2  1  1 2X  X  2  1  0  1  1  X 3X+2  1 X+2  2 2X X+2  2  0  1  1 3X  1 2X+2  1  1  1 X+2 2X+2  1  1 3X+2  1  1
 0  1  0 2X 2X+3  3  1  X 3X 3X 3X+3 X+3  1  1 2X+2  1 3X+2  1  1 3X+2  3  X 3X+1  1 3X+3  2  1 2X+1  0  X  1  1 X+1  2 3X  2  1 X+2  1  1  0 2X+2  1  1 X+1  1  2 X+3 3X 2X  X  1 2X+2 X+2 3X+2 3X+2  1 2X 2X+2  1 3X  1 2X 2X+3 2X+2  1  1 X+3 X+1  1 2X+3  0
 0  0  1 3X+1 X+1 2X 3X+1 3X 2X+3  1  3  X X+2 2X+1 3X X+2  1 X+1 3X+2 3X+1 2X+1  2 X+2 2X+1 X+1  1 3X 2X+2  1 2X  2  3  0 2X+2 X+3  1 3X X+2 X+3  X 2X+3  1  1 2X 2X+1 3X+3 X+1 2X  1  1  1  0  1  1  1  1 3X 2X+2 3X+2 3X+2 X+2 2X+3  X 3X X+3  X 2X+2 3X+3 2X+2  0 2X+3  0

generates a code of length 72 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+142x^68+620x^69+700x^70+652x^71+622x^72+332x^73+217x^74+272x^75+164x^76+144x^77+98x^78+92x^79+32x^80+6x^84+1x^88+1x^90

The gray image is a code over GF(2) with n=576, k=12 and d=272.
This code was found by Heurico 1.16 in 0.328 seconds.