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 X^3  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^3+X^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 X^3  X X^2 X^2 X^2+X X^2+X X^2 X^2 X^2+X X^3 X^2+X X^3 X^3+X^2 X^3+X X^3+X^2+X X^3+X^2 X^3 X^3+X^2+X X^3+X X^3 X^3+X^2 X^3+X X^3+X^2+X X^3+X^2 X^3 X^3+X^2+X X^3+X X^3 X^3 X^3+X X^3+X X^3+X^2+X X^3+X^2+X X^3+X^2 X^3+X^2 X^3+X X^3+X X^3 X^3 X^3+X^2 X^3+X^2 X^3+X^2+X X^3+X^2+X  X X^2+X  X  X  X  X X^3+X^2+X  X X^2  0 X^3+X^2 X^3  0
 0  0  X  X X^3+X^2 X^2+X X^3+X^2+X X^2 X^2 X^2+X X^3+X X^3 X^3+X^2+X X^3 X^3+X X^3+X^2 X^3 X^3+X^2+X X^3+X X^3+X^2 X^3+X X^2 X^2+X  X X^3 X^3+X^2  X X^3+X^2+X  0  0 X^2+X  X  X X^2 X^3 X^3+X X^3+X X^3 X^2 X^2+X X^3+X^2+X X^3+X^2 X^2 X^2+X X^2+X X^2 X^3 X^3+X  X  0  0 X^3+X^2+X  X X^2 X^2+X  0 X^3+X^2  X X^3+X^2+X X^3 X^3+X^2  X  0 X^3+X^2+X X^3+X X^3+X^2  X  0  0  X X^3+X^2+X X^3+X^2 X^2 X^2+X X^3+X X^2+X X^2+X X^3+X  0

generates a code of length 79 over Z2[X]/(X^4) 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 linear code over GF(2) with n=632, k=10 and d=304.
This code was found by Heurico 1.16 in 0.328 seconds.