The generator matrix

 1  0  0  1  1  1 X^3 X^2  1  1  0  1  1  0  X  1  1 X^2+X  1  1 X^3+X X^3+X^2+X X^3+X  1 X^2  1  1 X^2+X  1  1  X  1  1  1  1  1 X^2  1
 0  1  0  0 X^3+X^2+1 X^3+X^2+1  1  X X^3 X^2+1  1 X^3 X^2+1  1  1  X X+1  1 X^3+X^2+X X^3+X^2+X+1 X^3+X^2+X X^2  1 X^3+X^2+X+1  1 X^3+X+1  1  1 X^3+X^2+X+1  X  1 X^3+X^2 X^3+X^2 X^2+X  X X^3+X^2  1 X^2+X
 0  0  1 X+1 X^2+X+1 X^2 X^2+X+1  1 X^3+X^2+X X^3+1 X^3+1 X^3+X^2+1 X^3+X X^2+X  X  X X^3  1 X^2+X+1 X^3+X+1  1  1  0 X^3+X^2+1  1 X^3+X^2+X X^2 X^3+X+1 X^3+1 X^3+X^2+1  0 X^2+X+1  1 X^2+1  1 X^2+1 X^3+X+1 X^3+X
 0  0  0 X^3 X^3  0 X^3 X^3  0 X^3 X^3 X^3  0  0 X^3 X^3 X^3  0  0  0 X^3 X^3 X^3  0  0 X^3 X^3  0 X^3  0  0  0 X^3 X^3  0  0  0  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+187x^34+804x^35+1204x^36+1338x^37+1490x^38+1278x^39+776x^40+650x^41+327x^42+58x^43+28x^44+28x^45+12x^46+4x^47+7x^48

The gray image is a linear code over GF(2) with n=304, k=13 and d=136.
This code was found by Heurico 1.16 in 0.75 seconds.