The generator matrix

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

generates a code of length 78 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 75.

Homogenous weight enumerator: w(x)=1x^0+378x^75+242x^76+344x^77+258x^78+340x^79+155x^80+188x^81+46x^82+74x^83+12x^87+4x^89+4x^91+2x^112

The gray image is a code over GF(2) with n=624, k=11 and d=300.
This code was found by Heurico 1.16 in 11.3 seconds.