The generator matrix

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

generates a code of length 28 over Z2[X]/(X^4) who´s minimum homogenous weight is 20.

Homogenous weight enumerator: w(x)=1x^0+29x^20+66x^24+128x^26+256x^27+1094x^28+256x^29+128x^30+60x^32+25x^36+4x^44+1x^48

The gray image is a linear code over GF(2) with n=224, k=11 and d=80.
This code was found by Heurico 1.16 in 0.063 seconds.