The generator matrix

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

generates a code of length 40 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 37.

Homogenous weight enumerator: w(x)=1x^0+416x^37+690x^38+860x^39+653x^40+514x^41+396x^42+264x^43+136x^44+138x^45+9x^46+16x^47+1x^48+1x^50+1x^52

The gray image is a code over GF(2) with n=320, k=12 and d=148.
This code was found by Heurico 1.16 in 0.11 seconds.