The generator matrix

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

generates a code of length 60 over Z2[X]/(X^4) who´s minimum homogenous weight is 53.

Homogenous weight enumerator: w(x)=1x^0+580x^53+1795x^54+4272x^55+7078x^56+10726x^57+14137x^58+17330x^59+18962x^60+17844x^61+14532x^62+10726x^63+6712x^64+3740x^65+1525x^66+664x^67+256x^68+144x^69+11x^70+14x^71+14x^72+6x^73+2x^75+1x^88

The gray image is a linear code over GF(2) with n=480, k=17 and d=212.
This code was found by Heurico 1.16 in 122 seconds.