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

generates a code of length 72 over Z2[X]/(X^4) who´s minimum homogenous weight is 69.

Homogenous weight enumerator: w(x)=1x^0+16x^69+112x^70+244x^71+291x^72+232x^73+95x^74+20x^75+12x^76+1x^130

The gray image is a linear code over GF(2) with n=576, k=10 and d=276.
This code was found by Heurico 1.16 in 0.5 seconds.