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

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+30x^77+207x^78+15x^80+2x^93+1x^126

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 14.1 seconds.