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

generates a code of length 72 over Z2[X]/(X^4) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+18x^68+32x^69+72x^70+96x^71+586x^72+96x^73+72x^74+32x^75+17x^76+1x^80+1x^140

The gray image is a linear code over GF(2) with n=576, k=10 and d=272.
This code was found by Heurico 1.16 in 0.375 seconds.