The generator matrix

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

generates a code of length 38 over Z2[X]/(X^4) who´s minimum homogenous weight is 36.

Homogenous weight enumerator: w(x)=1x^0+436x^36+168x^38+401x^40+8x^42+8x^44+1x^48+1x^56

The gray image is a linear code over GF(2) with n=304, k=10 and d=144.
This code was found by Heurico 1.16 in 2.8 seconds.