The generator matrix

 1  0  1  1  1 X^2+X  1  1 X^3+X^2  1  1 X^3+X  1  1  0  1  1 X^2+X  1  1 X^3+X^2  1  1 X^3+X  1  1  0  1  1 X^2+X  1  1 X^3+X^2  1 X^3+X  1  1  1  1  1  1  1  1  1  0 X^2+X X^3+X^2 X^3+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
 0  1 X+1 X^2+X X^2+1  1 X^3+X^2+X+1 X^3+X^2  1 X^3+X X^3+1  1  0 X+1  1 X^2+X X^2+1  1 X^3+X^2 X^3+X^2+X+1  1 X^3+X X^3+1  1  0 X+1  1 X^2+X X^2+1  1 X^3+X^2 X^3+X^2+X+1  1 X^3+X  1 X^3+1 X^3+X^2+1 X+1  1 X^3+X^2+X+1  0 X^2+X X^3+X^2 X^3+X  1  1  1  1 X^2+1 X^3+X+1 X^2+X+1 X^3+1 X^2+1 X+1 X^3+1 X^3+X^2+X+1 X+1 X^2+1 X^3+X^2+1 X^3+X+1 X^3+X^2+X+1 X^2+X+1 X^3+1  1 X^3 X^2 X^3+X^2+X  X  0 X^3+X^2 X^2+X X^3+X^2+X
 0  0 X^3  0  0  0  0 X^3 X^3 X^3 X^3 X^3  0  0  0  0  0  0 X^3 X^3 X^3 X^3 X^3 X^3  0  0  0 X^3 X^3 X^3  0 X^3  0 X^3 X^3  0 X^3  0  0 X^3 X^3  0 X^3  0 X^3  0 X^3  0  0 X^3 X^3  0 X^3 X^3  0  0  0  0 X^3 X^3  0  0 X^3 X^3 X^3 X^3 X^3  0 X^3  0  0  0
 0  0  0 X^3  0 X^3 X^3 X^3 X^3  0 X^3  0 X^3  0 X^3  0 X^3  0  0 X^3  0 X^3  0 X^3  0  0 X^3 X^3 X^3  0 X^3  0  0  0 X^3 X^3  0 X^3  0 X^3 X^3  0  0 X^3 X^3  0  0 X^3  0  0  0 X^3 X^3 X^3  0 X^3 X^3 X^3  0 X^3  0  0 X^3  0  0 X^3  0 X^3 X^3 X^3 X^3 X^3
 0  0  0  0 X^3  0 X^3 X^3 X^3 X^3  0 X^3  0 X^3  0  0 X^3  0 X^3  0 X^3 X^3  0 X^3 X^3  0 X^3  0 X^3  0 X^3 X^3 X^3  0  0  0 X^3  0  0 X^3  0 X^3  0 X^3  0 X^3  0 X^3  0 X^3  0 X^3  0 X^3 X^3  0 X^3  0  0  0  0 X^3 X^3 X^3  0  0 X^3  0 X^3  0 X^3  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+244x^68+64x^69+240x^70+64x^71+820x^72+64x^73+240x^74+64x^75+244x^76+1x^80+1x^96+1x^112

The gray image is a linear code over GF(2) with n=576, k=11 and d=272.
This code was found by Heurico 1.16 in 0.297 seconds.