The generator matrix

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

generates a code of length 22 over Z2[X]/(X^4) who´s minimum homogenous weight is 18.

Homogenous weight enumerator: w(x)=1x^0+52x^18+128x^20+256x^21+157x^22+256x^23+122x^24+43x^26+4x^28+3x^30+1x^32+1x^34

The gray image is a linear code over GF(2) with n=176, k=10 and d=72.
This code was found by Heurico 1.16 in 0.094 seconds.