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

generates a code of length 78 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 76.

Homogenous weight enumerator: w(x)=1x^0+198x^76+128x^77+384x^78+128x^79+175x^80+9x^84+1x^148

The gray image is a code over GF(2) with n=624, k=10 and d=304.
This code was found by Heurico 1.16 in 4.64 seconds.