The generator matrix

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

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+78x^34+528x^35+1521x^36+2230x^37+4333x^38+4770x^39+6130x^40+4596x^41+4338x^42+2178x^43+1297x^44+446x^45+193x^46+86x^47+27x^48+8x^49+2x^50+6x^51

The gray image is a code over GF(2) with n=320, k=15 and d=136.
This code was found by Heurico 1.16 in 5.76 seconds.