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

generates a code of length 39 over Z2[X]/(X^4) who´s minimum homogenous weight is 34.

Homogenous weight enumerator: w(x)=1x^0+101x^34+16x^35+126x^36+192x^37+427x^38+352x^39+421x^40+192x^41+93x^42+16x^43+85x^44+17x^46+5x^48+2x^50+1x^52+1x^64

The gray image is a linear code over GF(2) with n=312, k=11 and d=136.
This code was found by Heurico 1.16 in 1.34 seconds.