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  X  X  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  X X^2  X X^2
 0 X^2+2  0 X^2  0  0 X^2 X^2+2  0  0 X^2 X^2+2  0  0 X^2 X^2+2  0  0 X^2 X^2+2  0  0 X^2 X^2+2  0  0  0  2  2 X^2+2  0 X^2 X^2 X^2+2 X^2 X^2+2 X^2 X^2+2  2  2  2  2  2 X^2+2  2 X^2  2 X^2  2 X^2+2  2 X^2  2  2 X^2+2  2  2 X^2 X^2+2 X^2 X^2+2  2  2  2  2 X^2+2 X^2  0  2 X^2 X^2+2  2 X^2+2  2 X^2 X^2 X^2+2 X^2+2 X^2
 0  0 X^2+2 X^2  0 X^2+2 X^2  0  0 X^2+2 X^2  0  0 X^2+2 X^2  0  2 X^2 X^2+2  2  2 X^2 X^2+2  2  2 X^2  2 X^2 X^2 X^2 X^2  2 X^2+2 X^2+2 X^2+2  2  2  2 X^2  2 X^2  2 X^2 X^2+2  2  2 X^2  2  2 X^2 X^2+2  0  0 X^2+2 X^2  0 X^2  0 X^2+2  0 X^2+2  0 X^2+2  0 X^2+2 X^2  0  0 X^2 X^2  0  2 X^2+2  0  0 X^2+2  2  0 X^2
 0  0  0  2  0  0  2  0  2  2  0  2  2  2  0  2  2  2  2  2  2  2  2  2  0  0  0  0  0  0  0  0  0  0  0  0  0  0  2  0  0  0  0  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  0  0  0  0  0  0  0  0  0  2  0  0  0  2  2  2  0  2  0  0
 0  0  0  0  2  2  2  2  2  2  0  2  0  0  2  0  0  0  0  0  2  2  2  2  2  2  0  0  2  2  0  0  2  0  0  0  2  2  2  0  0  2  2  2  2  2  2  0  0  0  0  0  0  2  2  2  0  2  0  2  2  2  0  0  2  0  0  0  0  0  0  2  0  2  0  0  0  2  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+100x^76+64x^77+224x^78+256x^79+234x^80+64x^81+48x^82+32x^84+1x^144

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 71 seconds.