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

generates a code of length 39 over Z2[X]/(X^4) who´s minimum homogenous weight is 35.

Homogenous weight enumerator: w(x)=1x^0+122x^35+153x^36+210x^37+369x^38+388x^39+370x^40+196x^41+94x^42+78x^43+32x^44+22x^45+1x^46+4x^47+3x^48+4x^49+1x^60

The gray image is a linear code over GF(2) with n=312, k=11 and d=140.
This code was found by Heurico 1.16 in 1.45 seconds.