The generator matrix

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

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+272x^36+152x^37+312x^38+80x^39+90x^40+24x^41+86x^42+4x^44+1x^48+2x^50

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 12.9 seconds.