The generator matrix

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

generates a code of length 38 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 36.

Homogenous weight enumerator: w(x)=1x^0+272x^36+152x^37+312x^38+80x^39+90x^40+24x^41+86x^42+4x^44+1x^48+2x^50

The gray image is a code over GF(2) with n=304, k=10 and d=144.
This code was found by Heurico 1.16 in 12.8 seconds.