The generator matrix

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

generates a code of length 40 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 34.

Homogenous weight enumerator: w(x)=1x^0+123x^34+1240x^35+2672x^36+4808x^37+7226x^38+10964x^39+11257x^40+11408x^41+7276x^42+4750x^43+2381x^44+1024x^45+270x^46+98x^47+22x^48+8x^49+1x^50+2x^51+3x^52+2x^55

The gray image is a code over GF(2) with n=320, k=16 and d=136.
This code was found by Heurico 1.16 in 21.1 seconds.