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

generates a code of length 38 over Z2[X]/(X^4) who´s minimum homogenous weight is 34.

Homogenous weight enumerator: w(x)=1x^0+57x^34+16x^35+50x^36+240x^37+307x^38+240x^39+39x^40+16x^41+47x^42+5x^44+5x^46+1x^68

The gray image is a linear code over GF(2) with n=304, k=10 and d=136.
This code was found by Heurico 1.16 in 2.17 seconds.