The generator matrix

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

generates a code of length 38 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 36.

Homogenous weight enumerator: w(x)=1x^0+344x^36+352x^38+317x^40+6x^44+1x^48+2x^52+1x^56

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 32.7 seconds.