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

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+22x^76+280x^77+38x^78+384x^79+39x^80+192x^81+23x^82+32x^83+2x^84+8x^85+2x^86+1x^130

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 0.344 seconds.