The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  0  1  1  1  0  X  1  1  1  1 X^2+2  2  1  1  1  X  1  1  1  1  1
 0  X  0  X  2  0 X^2+X X^2+X+2  0  2 X+2 X+2  0 X^2+X+2 X^2+2  X X^2+2  X X^2+X+2 X^2+2 X+2  X X+2 X+2 X^2+X X^2+2  X  X  2  0 X^2 X^2+2 X^2+2 X^2+2 X^2+X  2 X^2+X X^2+X
 0  0  X  X  0 X^2+X+2 X^2+X  2 X^2 X^2+X+2 X^2+X+2 X^2 X^2+2 X^2  X  X X^2+X+2 X^2+X+2 X+2 X^2+X+2  0  2 X^2+2  2 X+2  0  0  X  X X+2 X^2+X X^2+X+2  X X^2  2 X^2+X X^2+X  X
 0  0  0 X^2 X^2+2 X^2  2 X^2 X^2  0 X^2 X^2+2  0  0 X^2+2  2 X^2  0 X^2 X^2+2 X^2  0 X^2  2  2 X^2+2 X^2+2 X^2+2 X^2 X^2+2  2  0 X^2  2  2  2 X^2+2  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+336x^34+48x^35+692x^36+464x^37+1146x^38+464x^39+548x^40+48x^41+238x^42+100x^44+6x^46+2x^48+2x^50+1x^64

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 31.2 seconds.