The generator matrix

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

generates a code of length 60 over Z2[X]/(X^4) who´s minimum homogenous weight is 55.

Homogenous weight enumerator: w(x)=1x^0+428x^55+1009x^56+1764x^57+2082x^58+2114x^59+2297x^60+1908x^61+1793x^62+1324x^63+765x^64+524x^65+192x^66+86x^67+37x^68+24x^69+13x^70+16x^71+3x^72+4x^73

The gray image is a linear code over GF(2) with n=480, k=14 and d=220.
This code was found by Heurico 1.16 in 10.3 seconds.