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

generates a code of length 78 over Z4[X]/(X^2+2) who´s minimum homogenous weight is 77.

Homogenous weight enumerator: w(x)=1x^0+56x^77+143x^78+32x^79+15x^80+8x^85+1x^94

The gray image is a code over GF(2) with n=624, k=8 and d=308.
This code was found by Heurico 1.16 in 2.47 seconds.