The generator matrix

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

generates a code of length 11 over F4[X]/(X^2) who´s minimum homogenous weight is 20.

Homogenous weight enumerator: w(x)=1x^0+165x^20+780x^24+1536x^27+3951x^28+3072x^30+9216x^31+11523x^32+9216x^34+13824x^35+11523x^36+528x^40+201x^44

The gray image is a linear code over GF(4) with n=44, k=8 and d=20.
This code was found by Heurico 1.16 in 1.47 seconds.