The generator matrix

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

generates a code of length 77 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 74.

Homogenous weight enumerator: w(x)=1x^0+377x^74+284x^75+299x^76+284x^77+254x^78+202x^79+215x^80+24x^81+80x^82+6x^83+12x^84+8x^88+1x^110+1x^112

The gray image is a code over GF(2) with n=616, k=11 and d=296.
This code was found by Heurico 1.16 in 0.359 seconds.