The generator matrix

 1  0  0  1  1  1  1  1  1  1  1
 0  1  0  1  a a^2  0  1 a^2*X+a a^2*X+1 a*X+1
 0  0  1 a^2  a  1  a  1 a^2  a a*X+a
 0  0  0  X  0  X  0  X a^2*X a^2*X a^2*X
 0  0  0  0  X a^2*X a*X  X  0 a^2*X  X

generates a code of length 11 over F4[X]/(X^2) who´s minimum homogenous weight is 22.

Homogenous weight enumerator: w(x)=1x^0+180x^22+501x^24+720x^26+1536x^27+3189x^28+4824x^30+9216x^31+9237x^32+10512x^34+13824x^35+9402x^36+2196x^38+165x^40+33x^44

The gray image is a linear code over GF(4) with n=44, k=8 and d=22.
This code was found by Heurico 1.16 in 1.25 seconds.