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  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^3  1  1  1  1  1
 0  X  0 X^2+X X^2 X^3+X^2+X X^3+X^2  X  0 X^2+X  0 X^3+X^2+X X^2  X X^3+X^2  X  0 X^2+X  0 X^3+X^2+X X^2  X X^3+X^2  X  0 X^2+X X^3+X^2  X  0 X^3+X^2+X X^2  X X^3 X^3+X^2+X X^2 X^3+X X^3 X^2+X X^3+X^2 X^3+X X^3 X^3+X^2+X X^2 X^3+X X^3 X^2+X X^3+X^2 X^3+X X^3 X^3+X^2+X X^2 X^3+X X^3 X^2+X X^3+X^2 X^3+X X^3 X^3+X^2+X X^2 X^3+X X^3 X^2+X X^3+X^2 X^3+X  0 X^2+X X^3+X^2 X^3+X X^3 X^3+X^2+X X^3+X^2+X  X X^2 X^3+X^2+X  X  X X^3
 0  0 X^3+X^2  0 X^3+X^2 X^2  0 X^2 X^3 X^3 X^2 X^3+X^2 X^2 X^3+X^2 X^3 X^3  0  0 X^3+X^2 X^2 X^2 X^3+X^2 X^3 X^3 X^3 X^3  0  0 X^2 X^3+X^2 X^3+X^2 X^2 X^3 X^3 X^3 X^3 X^2 X^3+X^2 X^2 X^3+X^2  0  0  0  0 X^3+X^2 X^2 X^3+X^2 X^2 X^3 X^3 X^3 X^3 X^2 X^3+X^2 X^3+X^2 X^3+X^2  0  0  0  0 X^3+X^2 X^2 X^2 X^2  0  0  0 X^3 X^2 X^2 X^2 X^2 X^3+X^2 X^3+X^2 X^2 X^2 X^2
 0  0  0 X^3 X^3  0 X^3 X^3  0 X^3  0  0 X^3 X^3 X^3  0 X^3  0 X^3 X^3  0  0  0 X^3 X^3  0  0 X^3 X^3 X^3  0  0  0  0 X^3 X^3  0  0 X^3 X^3  0  0 X^3 X^3  0  0 X^3 X^3 X^3 X^3  0  0 X^3 X^3  0  0 X^3 X^3  0  0 X^3 X^3  0  0  0  0 X^3 X^3  0  0 X^3  0  0 X^3 X^3  0  0

generates a code of length 77 over Z2[X]/(X^4) who´s minimum homogenous weight is 74.

Homogenous weight enumerator: w(x)=1x^0+51x^74+42x^75+270x^76+304x^77+267x^78+36x^79+47x^80+1x^82+2x^83+2x^84+1x^150

The gray image is a linear code over GF(2) with n=616, k=10 and d=296.
This code was found by Heurico 1.16 in 0.641 seconds.