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

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

Homogenous weight enumerator: w(x)=1x^0+404x^76+48x^77+554x^78+288x^79+655x^80+400x^81+668x^82+192x^83+378x^84+96x^85+238x^86+114x^88+40x^90+14x^92+4x^94+1x^96+1x^128

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 57.6 seconds.