The generator matrix

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

generates a code of length 60 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 56.

Homogenous weight enumerator: w(x)=1x^0+88x^56+592x^57+560x^58+890x^59+508x^60+476x^61+224x^62+328x^63+134x^64+136x^65+70x^66+70x^67+12x^68+4x^69+2x^70+1x^72

The gray image is a code over GF(2) with n=480, k=12 and d=224.
This code was found by Heurico 1.16 in 0.219 seconds.