The generator matrix

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

generates a code of length 76 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 69.

Homogenous weight enumerator: w(x)=1x^0+68x^69+634x^70+1458x^71+2054x^72+3212x^73+3291x^74+3988x^75+3949x^76+4144x^77+3199x^78+2608x^79+1691x^80+1276x^81+563x^82+308x^83+149x^84+60x^85+52x^86+22x^87+26x^88+8x^89+4x^90+1x^94+2x^96

The gray image is a code over GF(2) with n=608, k=15 and d=276.
This code was found by Heurico 1.16 in 12.9 seconds.