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

generates a code of length 81 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 76.

Homogenous weight enumerator: w(x)=1x^0+409x^76+8x^77+408x^78+288x^79+625x^80+688x^81+600x^82+288x^83+405x^84+8x^85+280x^86+54x^88+24x^90+9x^92+1x^148

The gray image is a code over GF(2) with n=648, k=12 and d=304.
This code was found by Heurico 1.16 in 26.2 seconds.