The generator matrix

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

generates a code of length 62 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 55.

Homogenous weight enumerator: w(x)=1x^0+168x^55+1044x^56+2294x^57+4034x^58+5270x^59+7601x^60+7584x^61+9331x^62+8518x^63+7545x^64+4982x^65+3645x^66+1808x^67+1005x^68+414x^69+169x^70+54x^71+40x^72+4x^73+13x^74+6x^75+4x^76+2x^77

The gray image is a code over GF(2) with n=496, k=16 and d=220.
This code was found by Heurico 1.16 in 32.5 seconds.