The generator matrix

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

generates a code of length 77 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 75.

Homogenous weight enumerator: w(x)=1x^0+148x^75+94x^76+536x^77+94x^78+148x^79+1x^90+1x^96+1x^122

The gray image is a code over GF(2) with n=616, k=10 and d=300.
This code was found by Heurico 1.16 in 0.36 seconds.