The generator matrix

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

generates a code of length 67 over Z2[X]/(X^4) who´s minimum homogenous weight is 60.

Homogenous weight enumerator: w(x)=1x^0+176x^60+1194x^61+2490x^62+3980x^63+5683x^64+6752x^65+8235x^66+8772x^67+8318x^68+7228x^69+5273x^70+3412x^71+2144x^72+1028x^73+522x^74+218x^75+38x^76+54x^77+15x^78+1x^82+2x^83

The gray image is a linear code over GF(2) with n=536, k=16 and d=240.
This code was found by Heurico 1.16 in 33.6 seconds.