The generator matrix

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

generates a code of length 35 over Z2[X]/(X^4) who´s minimum homogenous weight is 29.

Homogenous weight enumerator: w(x)=1x^0+272x^29+1652x^30+4422x^31+8926x^32+15066x^33+22700x^34+24624x^35+23033x^36+15552x^37+9022x^38+3822x^39+1379x^40+466x^41+80x^42+28x^43+19x^44+4x^45+2x^46+2x^48

The gray image is a linear code over GF(2) with n=280, k=17 and d=116.
This code was found by Heurico 1.16 in 64.7 seconds.