The generator matrix

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

generates a code of length 38 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 34.

Homogenous weight enumerator: w(x)=1x^0+44x^34+208x^35+251x^36+340x^37+377x^38+336x^39+242x^40+200x^41+36x^42+1x^44+4x^45+6x^46+1x^54+1x^56

The gray image is a code over GF(2) with n=304, k=11 and d=136.
This code was found by Heurico 1.16 in 0.062 seconds.