The generator matrix

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

generates a code of length 26 over Z2[X]/(X^4) who´s minimum homogenous weight is 22.

Homogenous weight enumerator: w(x)=1x^0+184x^22+144x^23+687x^24+628x^25+891x^26+640x^27+571x^28+104x^29+195x^30+16x^31+20x^32+4x^33+9x^34+1x^36+1x^38

The gray image is a linear code over GF(2) with n=208, k=12 and d=88.
This code was found by Heurico 1.16 in 0.641 seconds.