The generator matrix

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

generates a code of length 38 over Z4[X]/(X^2+2) who´s minimum homogenous weight is 34.

Homogenous weight enumerator: w(x)=1x^0+374x^34+128x^35+762x^36+400x^37+932x^38+368x^39+619x^40+112x^41+272x^42+16x^43+89x^44+20x^46+2x^50+1x^52

The gray image is a code over GF(2) with n=304, k=12 and d=136.
This code was found by Heurico 1.16 in 51.2 seconds.