The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 X^2  1  1  1  1  1  X  1
 0 X^2+2  0 X^2  0  2 X^2+2 X^2+2  0  2 X^2 X^2+2  0  0 X^2+2 X^2+2  0  2 X^2+2 X^2  2  2  0 X^2 X^2+2  2  0 X^2 X^2+2 X^2  2  0 X^2+2 X^2+2 X^2  0  2  0
 0  0 X^2+2 X^2  0 X^2 X^2  2  0 X^2 X^2  0  0 X^2+2 X^2  2  2  2  0  0 X^2 X^2+2 X^2 X^2+2 X^2+2  2  2 X^2+2 X^2+2  0  0 X^2  2  0  2 X^2+2 X^2  0
 0  0  0  2  0  0  2  0  2  2  0  2  2  2  0  2  2  0  0  2  0  2  0  2  0  0  2  0  2  0  0  2  2  2  0  0  2  0
 0  0  0  0  2  2  2  2  2  2  0  2  0  0  2  0  0  0  2  2  0  2  2  2  2  2  2  0  0  2  0  0  2  0  0  2  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+39x^34+114x^36+64x^37+636x^38+64x^39+52x^40+23x^42+24x^44+6x^46+1x^72

The gray image is a code over GF(2) with n=304, k=10 and d=136.
This code was found by Heurico 1.16 in 0.078 seconds.