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

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+69x^54+82x^56+64x^57+88x^58+384x^59+728x^60+320x^61+178x^62+36x^64+48x^66+48x^68+1x^70+1x^112

The gray image is a linear code over GF(2) with n=480, k=11 and d=216.
This code was found by Heurico 1.16 in 0.297 seconds.