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  X  1  1  X  1  1
 0 X^3+X^2  0 X^3+X^2  0 X^3+X^2  0 X^3+X^2  0 X^3+X^2  0 X^3+X^2  0 X^3+X^2  0 X^3+X^2  0 X^3+X^2  0 X^3+X^2  0 X^3+X^2  0 X^3+X^2 X^3 X^2 X^3 X^2 X^3 X^2 X^3 X^2  0 X^3+X^2 X^3 X^2 X^3 X^2 X^3 X^2  0 X^3+X^2 X^3 X^2 X^3 X^2 X^3 X^2 X^3 X^2 X^3 X^3+X^2  0 X^2 X^3 X^2  0 X^3 X^3+X^2 X^2 X^3 X^2 X^3 X^2  0 X^3+X^2 X^3+X^2  0 X^3+X^2 X^3+X^2  0  0
 0  0 X^3  0  0  0 X^3  0  0  0  0  0 X^3  0 X^3  0  0 X^3  0 X^3  0 X^3  0 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3 X^3  0  0  0  0 X^3 X^3 X^3 X^3  0  0  0  0  0 X^3  0 X^3 X^3  0 X^3  0 X^3  0 X^3 X^3 X^3  0  0  0  0  0  0  0  0  0 X^3 X^3
 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 X^3  0 X^3  0 X^3  0 X^3  0  0 X^3 X^3 X^3  0  0 X^3  0  0 X^3 X^3 X^3  0  0  0 X^3 X^3  0 X^3 X^3  0  0 X^3 X^3  0 X^3  0 X^3  0  0  0  0  0  0 X^3 X^3 X^3 X^3
 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 X^3  0  0 X^3  0 X^3 X^3  0 X^3  0  0 X^3  0 X^3  0 X^3  0 X^3  0 X^3  0 X^3  0  0 X^3 X^3 X^3  0 X^3 X^3 X^3  0  0  0 X^3  0 X^3  0 X^3 X^3 X^3  0  0 X^3  0  0  0  0  0 X^3 X^3  0  0  0
 0  0  0  0  0 X^3  0 X^3 X^3 X^3 X^3  0 X^3  0 X^3 X^3  0  0 X^3 X^3 X^3 X^3  0  0  0  0 X^3 X^3 X^3 X^3  0  0  0  0  0  0  0  0  0  0 X^3 X^3  0 X^3 X^3  0  0 X^3 X^3 X^3 X^3  0 X^3 X^3 X^3 X^3  0  0 X^3  0 X^3 X^3 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+24x^68+80x^70+814x^72+80x^74+23x^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 0.391 seconds.