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  X  1  X  1  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  X  X X^2 X^2 X^2  0 X^2  X X^2  1
 0 X^3+X^2  0 X^2  0  0 X^2 X^3+X^2  0  0 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  0 X^3+X^2  0 X^3  0 X^3 X^2 X^3+X^2 X^3  0 X^3 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^2 X^2 X^3+X^2 X^3  0 X^3 X^2 X^2  0
 0  0 X^3+X^2 X^2  0 X^3+X^2 X^2  0  0 X^3+X^2 X^2  0  0 X^3+X^2 X^2  0 X^3 X^2 X^3+X^2 X^3 X^3 X^3+X^2 X^2 X^3 X^3 X^2 X^2 X^2 X^3+X^2 X^3 X^2 X^3 X^3+X^2 X^3+X^2 X^2 X^3 X^3 X^3+X^2 X^2 X^3 X^3 X^2 X^2  0 X^3 X^3+X^2 X^2 X^3 X^3 X^2 X^2  0  0 X^3+X^2 X^3+X^2 X^3  0 X^2 X^3+X^2  0  0 X^3+X^2 X^3+X^2 X^3  0 X^3+X^2 X^2  0 X^2  0 X^2  0 X^2 X^2  0 X^2 X^3+X^2  0
 0  0  0 X^3  0  0 X^3  0 X^3 X^3  0 X^3 X^3 X^3  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0  0  0  0  0  0 X^3  0 X^3  0  0  0  0  0  0  0  0  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0 X^3 X^3 X^3  0 X^3 X^3  0 X^3  0  0  0  0 X^3  0  0  0  0 X^3  0 X^3  0 X^3 X^3  0
 0  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3  0 X^3  0  0 X^3  0  0  0  0  0 X^3 X^3 X^3 X^3 X^3  0 X^3 X^3 X^3 X^3 X^3  0  0  0  0  0  0  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0  0  0  0  0 X^3  0  0 X^3  0 X^3 X^3 X^3  0 X^3 X^3  0  0 X^3  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3  0  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+42x^74+50x^75+187x^76+144x^77+213x^78+148x^79+121x^80+32x^81+63x^82+10x^83+11x^84+1x^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 0.515 seconds.