The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 X^2  1  1  1  1  1  0  1  1  1  1  1  X  1  1  X  1  X  1 X^3  X  1  X  1 X^3  1  X  1  0  1  1  0  X  X
 0  X  0  X X^3  0 X^2+X X^3+X^2+X  0 X^3 X^3+X X^3+X  0 X^3+X^2+X X^3+X^2  X X^3+X^2 X^2+X X^2+X X^3+X^2 X^3+X^2+X X^3+X X^3+X^2 X^2 X^3+X X^3+X^2 X^3 X^3+X  0 X^2 X^2+X X^2+X  X X^2  0 X^3+X^2+X X^3+X^2+X X^2  X  0 X^3 X^3+X X^3+X^2 X^3+X^2+X  0  X X^3+X^2 X^2 X^2+X X^3+X^2  0  X X^3 X^3+X^2+X  X X^3 X^2  X X^3 X^3+X^2
 0  0  X  X  0 X^3+X^2+X X^2+X X^3 X^2 X^3+X^2+X X^3+X^2+X X^2 X^3+X^2 X^2  X  X X^3+X^2+X X^3+X  0 X^3  X  0 X^3+X X^2  0 X^3  X X^2+X X^2+X X^3+X^2 X^3+X X^2  X X^3+X^2+X  X X^3+X^2+X X^2  0 X^2+X X^3 X^3+X^2+X X^3+X^2+X X^3+X^2 X^3+X^2 X^2 X^3  X X^2 X^3+X^2+X X^3+X  X X^3+X  X X^3 X^2+X X^3+X^2+X X^2+X X^3+X^2 X^3+X^2+X  X
 0  0  0 X^2 X^3+X^2 X^2 X^3 X^2 X^2  0 X^2 X^3+X^2  0  0 X^3+X^2 X^3 X^2 X^3+X^2  0 X^2  0 X^2  0  0 X^3 X^3 X^3+X^2 X^3 X^3+X^2 X^3+X^2 X^2 X^3+X^2 X^3+X^2 X^3 X^2  0 X^3 X^3+X^2 X^3+X^2 X^3 X^3 X^3 X^3  0 X^3+X^2 X^2 X^3+X^2 X^2 X^3 X^3+X^2  0 X^3+X^2  0 X^3  0 X^2  0  0 X^3 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+166x^55+209x^56+450x^57+501x^58+520x^59+604x^60+570x^61+382x^62+226x^63+142x^64+158x^65+53x^66+60x^67+27x^68+22x^69+4x^71+1x^92

The gray image is a linear code over GF(2) with n=480, k=12 and d=220.
This code was found by Heurico 1.16 in 0.578 seconds.