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

generates a code of length 76 over Z2[X]/(X^4) who´s minimum homogenous weight is 74.

Homogenous weight enumerator: w(x)=1x^0+232x^74+192x^75+216x^76+192x^77+144x^78+38x^80+8x^82+1x^128

The gray image is a linear code over GF(2) with n=608, k=10 and d=296.
This code was found by Heurico 1.16 in 8.14 seconds.