The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+72x^34+1098x^35+2840x^36+4732x^37+8168x^38+9702x^39+12350x^40+9776x^41+8460x^42+4520x^43+2410x^44+1006x^45+240x^46+134x^47+13x^48+4x^49+4x^50+2x^51+2x^52+2x^53

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 20.6 seconds.