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

generates a code of length 70 over Z2[X]/(X^4) who´s minimum homogenous weight is 64.

Homogenous weight enumerator: w(x)=1x^0+40x^64+56x^65+44x^66+98x^67+47x^68+502x^69+544x^70+488x^71+22x^72+88x^73+24x^74+18x^75+12x^76+22x^77+19x^78+4x^79+5x^80+4x^81+8x^82+1x^84+1x^126

The gray image is a linear code over GF(2) with n=560, k=11 and d=256.
This code was found by Heurico 1.16 in 0.454 seconds.