The generator matrix

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

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+59x^22+114x^23+166x^24+218x^25+399x^26+564x^27+818x^28+1152x^29+1202x^30+1144x^31+881x^32+596x^33+328x^34+204x^35+166x^36+80x^37+57x^38+22x^39+16x^40+2x^41+1x^42+2x^46

The gray image is a linear code over GF(2) with n=120, k=13 and d=44.
This code was found by Heurico 1.16 in 1.74 seconds.