The generator matrix

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

generates a code of length 35 over Z2[X]/(X^4) who´s minimum homogenous weight is 30.

Homogenous weight enumerator: w(x)=1x^0+784x^30+2354x^31+4606x^32+7982x^33+10725x^34+12444x^35+11158x^36+8004x^37+4506x^38+2050x^39+659x^40+174x^41+65x^42+16x^43+8x^44

The gray image is a linear code over GF(2) with n=280, k=16 and d=120.
This code was found by Heurico 1.16 in 24 seconds.