The generator matrix

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

generates a code of length 55 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 49.

Homogenous weight enumerator: w(x)=1x^0+564x^49+2035x^50+3798x^51+5200x^52+7710x^53+8467x^54+10070x^55+9081x^56+7448x^57+4777x^58+3552x^59+1808x^60+702x^61+205x^62+46x^63+28x^64+24x^65+12x^66+6x^67+2x^68

The gray image is a code over GF(2) with n=440, k=16 and d=196.
This code was found by Heurico 1.16 in 30.6 seconds.