The generator matrix

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

generates a code of length 23 over Z2[X]/(X^4) who´s minimum homogenous weight is 22.

Homogenous weight enumerator: w(x)=1x^0+6x^22+44x^23+7x^24+2x^26+4x^27

The gray image is a linear code over GF(2) with n=184, k=6 and d=88.
This code was found by Heurico 1.16 in 6.87e-008 seconds.