The generator matrix

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

generates a code of length 44 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 38.

Homogenous weight enumerator: w(x)=1x^0+222x^38+1384x^39+2901x^40+5304x^41+7288x^42+10420x^43+10613x^44+10480x^45+7358x^46+5208x^47+2593x^48+1264x^49+320x^50+108x^51+55x^52+8x^53+4x^54+3x^56+2x^60

The gray image is a code over GF(2) with n=352, k=16 and d=152.
This code was found by Heurico 1.16 in 22 seconds.