The generator matrix

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

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+110x^34+718x^35+1279x^36+2164x^37+2465x^38+3030x^39+2623x^40+2122x^41+917x^42+556x^43+251x^44+96x^45+27x^46+14x^47+4x^48+2x^49+1x^50+2x^51+2x^52

The gray image is a linear code over GF(2) with n=312, k=14 and d=136.
This code was found by Heurico 1.16 in 1.88 seconds.