The generator matrix

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

generates a code of length 23 over Z2[X]/(X^4) who´s minimum homogenous weight is 20.

Homogenous weight enumerator: w(x)=1x^0+11x^20+16x^22+192x^23+31x^24+4x^28+1x^44

The gray image is a linear code over GF(2) with n=184, k=8 and d=80.
This code was found by Heurico 1.16 in 6.87e-008 seconds.