The generator matrix

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

generates a code of length 62 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 55.

Homogenous weight enumerator: w(x)=1x^0+164x^55+1098x^56+2480x^57+3416x^58+5944x^59+6776x^60+8736x^61+8443x^62+8978x^63+6914x^64+5710x^65+3212x^66+2026x^67+910x^68+428x^69+171x^70+82x^71+29x^72+6x^73+6x^74+6x^75

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 35.6 seconds.