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

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+218x^76+128x^77+364x^78+128x^79+146x^80+36x^82+2x^84+1x^144

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 0.312 seconds.