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

generates a code of length 78 over Z2[X]/(X^4) who´s minimum homogenous weight is 74.

Homogenous weight enumerator: w(x)=1x^0+27x^74+90x^75+110x^76+172x^77+239x^78+160x^79+107x^80+84x^81+16x^82+6x^83+6x^84+5x^86+1x^130

The gray image is a linear code over GF(2) with n=624, k=10 and d=296.
This code was found by Heurico 1.16 in 0.406 seconds.