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  2  1  1  1  1  1  1  1  1  1  1  X  1  2  1
 0  X  0 X^2+X+2  2 X^2+X  2 X+2  0 X^2+X+2  0  X X^2+X+2  0  2  X  0 X^2+X+2 X+2  2  0  X X^2+X  2  0 X^2+X  X  2  2 X^2+X  0  X X^2+X X^2 X^2+X X^2+2 X+2  X X^2 X^2 X^2 X^2 X^2+X  X X^2+2 X^2+X X+2 X^2 X^2 X^2+X X^2+X+2  X X^2+2 X^2+X X+2 X^2 X^2+2 X^2 X^2+X+2 X^2+2 X^2+2  X X+2  2 X^2 X^2 X^2 X^2 X^2+2  0 X^2+2 X+2 X^2+X+2  2 X^2+X  X  X X+2
 0  0 X^2+2  0  0 X^2+2 X^2 X^2  0  0  0  0 X^2+2 X^2 X^2+2 X^2 X^2+2  2  2 X^2  2 X^2+2 X^2  2 X^2+2  2 X^2+2  2  2 X^2 X^2  2  0 X^2 X^2+2  0 X^2  0  2 X^2+2 X^2+2  2  0  0  2 X^2+2 X^2 X^2 X^2+2  2  2  2  0 X^2 X^2+2 X^2+2 X^2 X^2 X^2  0 X^2+2 X^2+2  2  0  0  2 X^2  2  2 X^2 X^2+2 X^2 X^2  0  2  2  0  0
 0  0  0 X^2+2 X^2 X^2+2 X^2  0  2 X^2 X^2+2  2 X^2 X^2+2  2  2 X^2 X^2+2  0  2 X^2  2 X^2  0 X^2+2 X^2  0  2 X^2+2 X^2+2  0  2  0  0  0  0 X^2 X^2 X^2+2  2 X^2 X^2  2 X^2+2  0  2 X^2+2 X^2+2 X^2+2  2  0 X^2+2 X^2+2  2 X^2+2  0  2  0  0  2 X^2+2 X^2 X^2  0 X^2  2 X^2  0 X^2 X^2  2  2 X^2  0  0 X^2 X^2  2

generates a code of length 78 over Z4[X]/(X^3+2,2X) 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 106 seconds.