The generator matrix

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

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+128x^34+72x^35+248x^36+316x^37+564x^38+324x^39+216x^40+44x^41+84x^42+4x^43+28x^44+8x^45+8x^46+2x^48+1x^64

The gray image is a linear code over GF(2) with n=304, k=11 and d=136.
This code was found by Heurico 1.16 in 0.266 seconds.