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

generates a code of length 60 over Z2[X]/(X^4) who´s minimum homogenous weight is 54.

Homogenous weight enumerator: w(x)=1x^0+914x^54+2252x^55+3907x^56+5588x^57+7237x^58+8404x^59+8776x^60+8706x^61+7707x^62+5504x^63+3524x^64+1648x^65+792x^66+364x^67+142x^68+42x^69+21x^70+4x^71+2x^72+1x^74

The gray image is a linear code over GF(2) with n=480, k=16 and d=216.
This code was found by Heurico 1.16 in 53.1 seconds.