The generator matrix

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

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+539x^64+1132x^65+2274x^66+3072x^67+3942x^68+3480x^69+4336x^70+3752x^71+3737x^72+2824x^73+1706x^74+872x^75+606x^76+168x^77+208x^78+40x^79+30x^80+12x^81+16x^82+8x^83+8x^84+4x^86+1x^88

The gray image is a linear code over GF(2) with n=560, k=15 and d=256.
This code was found by Heurico 1.16 in 567 seconds.