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

generates a code of length 78 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 77.

Homogenous weight enumerator: w(x)=1x^0+30x^77+206x^78+14x^80+2x^93+2x^94+1x^96

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 2.83 seconds.