The generator matrix

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

generates a code of length 59 over Z2[X]/(X^4) who´s minimum homogenous weight is 53.

Homogenous weight enumerator: w(x)=1x^0+786x^53+2494x^54+3512x^55+5554x^56+6974x^57+9006x^58+8900x^59+9326x^60+7146x^61+5562x^62+3126x^63+1797x^64+754x^65+379x^66+120x^67+46x^68+36x^69+6x^70+6x^71+4x^72+1x^74

The gray image is a linear code over GF(2) with n=472, k=16 and d=212.
This code was found by Heurico 1.16 in 47.3 seconds.