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

generates a code of length 79 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 76.

Homogenous weight enumerator: w(x)=1x^0+68x^76+24x^77+138x^78+592x^79+112x^80+24x^81+44x^82+18x^84+2x^86+1x^152

The gray image is a code over GF(2) with n=632, k=10 and d=304.
This code was found by Heurico 1.16 in 26.5 seconds.