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

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+196x^55+186x^56+468x^57+284x^58+870x^59+395x^60+692x^61+249x^62+346x^63+108x^64+132x^65+23x^66+82x^67+29x^68+20x^69+3x^70+10x^71+1x^72+1x^90

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 0.469 seconds.