The generator matrix

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

generates a code of length 72 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 66.

Homogenous weight enumerator: w(x)=1x^0+597x^66+1226x^67+2393x^68+2722x^69+3958x^70+3698x^71+4339x^72+3660x^73+3572x^74+2396x^75+2106x^76+970x^77+642x^78+242x^79+115x^80+40x^81+32x^82+22x^83+25x^84+6x^86+5x^88+1x^90

The gray image is a code over GF(2) with n=576, k=15 and d=264.
This code was found by Heurico 1.16 in 45.3 seconds.