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

generates a code of length 78 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 74.

Homogenous weight enumerator: w(x)=1x^0+114x^74+112x^75+329x^76+368x^77+384x^78+304x^79+209x^80+48x^81+62x^82+32x^83+44x^84+32x^85+8x^86+1x^148

The gray image is a code over GF(2) with n=624, k=11 and d=296.
This code was found by Heurico 1.16 in 108 seconds.