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 X^3  1  1  1  1  0  X X^3+X^2  X  0  X  X
 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 X^3 X^3+X^2+X X^2 X^3+X  0 X^2+X X^3+X X^3+X^2 X^2+X  0 X^3+X X^3+X^2  0 X^2+X X^3+X X^3+X^2 X^3 X^3+X^2+X X^2  X  0 X^2+X X^3+X^2 X^3+X X^2 X^3+X^2+X  0  X X^2+X X^3 X^3+X X^2  X X^3+X^2 X^3 X^3+X^2+X  0 X^3  0 X^3 X^2+X X^3+X^2+X X^2+X X^3+X^2+X X^3+X^2 X^3+X^2 X^2 X^2 X^3 X^3+X X^3+X  X  X  X X^2+X  X X^3+X  X X^2+X  0
 0  0 X^3  0  0  0 X^3  0  0  0  0 X^3  0  0 X^3  0  0 X^3 X^3 X^3 X^3  0 X^3 X^3 X^3 X^3 X^3  0 X^3 X^3  0 X^3  0  0  0  0  0 X^3 X^3 X^3  0  0  0  0 X^3  0 X^3 X^3 X^3 X^3 X^3 X^3  0 X^3 X^3  0  0 X^3 X^3  0  0  0 X^3 X^3  0  0  0  0  0 X^3 X^3  0
 0  0  0 X^3  0  0  0 X^3  0  0  0  0  0 X^3  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0  0 X^3 X^3  0 X^3  0 X^3  0  0 X^3 X^3 X^3  0 X^3  0 X^3 X^3 X^3 X^3  0  0 X^3 X^3  0 X^3  0 X^3  0  0 X^3  0 X^3 X^3  0  0  0  0  0  0  0  0  0 X^3 X^3  0  0
 0  0  0  0 X^3  0 X^3  0  0 X^3 X^3 X^3 X^3 X^3  0 X^3 X^3  0  0  0 X^3 X^3 X^3  0 X^3 X^3 X^3  0 X^3  0  0  0  0  0  0 X^3 X^3 X^3  0  0 X^3  0  0 X^3 X^3  0  0  0  0 X^3  0 X^3 X^3  0 X^3  0 X^3 X^3 X^3 X^3  0  0 X^3  0 X^3  0  0 X^3 X^3  0  0  0
 0  0  0  0  0 X^3  0 X^3 X^3 X^3  0 X^3 X^3  0 X^3 X^3  0  0 X^3  0  0 X^3 X^3 X^3 X^3  0 X^3 X^3  0  0  0 X^3 X^3  0  0  0  0 X^3 X^3  0  0  0  0 X^3  0 X^3  0 X^3  0 X^3 X^3  0 X^3 X^3 X^3 X^3 X^3 X^3  0  0  0 X^3  0  0 X^3  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+138x^68+192x^69+352x^70+192x^71+486x^72+192x^73+160x^74+64x^75+78x^76+128x^77+64x^78+1x^128

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 0.453 seconds.