The generator matrix

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

generates a code of length 71 over Z2[X]/(X^4) who´s minimum homogenous weight is 68.

Homogenous weight enumerator: w(x)=1x^0+435x^68+288x^69+456x^70+64x^71+290x^72+64x^73+176x^74+64x^75+145x^76+32x^77+24x^78+4x^80+2x^84+1x^88+2x^92

The gray image is a linear code over GF(2) with n=568, k=11 and d=272.
This code was found by Heurico 1.16 in 30.5 seconds.