The generator matrix

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

generates a code of length 30 over Z2[X]/(X^3) who´s minimum homogenous weight is 22.

Homogenous weight enumerator: w(x)=1x^0+88x^22+370x^23+1123x^24+1828x^25+3453x^26+4824x^27+7378x^28+8314x^29+10336x^30+8448x^31+8003x^32+4742x^33+3450x^34+1808x^35+830x^36+342x^37+144x^38+38x^39+9x^40+6x^41+1x^42

The gray image is a linear code over GF(2) with n=120, k=16 and d=44.
This code was found by Heurico 1.13 in 17.5 seconds.