The generator matrix

 1  1  1  1  1  1  1  1  1  1  1
 0  X  0  0  0  0  X  0  X a*X a*X
 0  0  X  0  0  0 a^2*X  X a^2*X  0 a^2*X
 0  0  0  X  0  0  X a^2*X a^2*X  X a*X
 0  0  0  0  X  0 a^2*X  X a*X a*X  0
 0  0  0  0  0  X a*X  X a^2*X a^2*X  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+396x^24+495x^28+1155x^32+12288x^33+1155x^36+528x^40+201x^44

The gray image is a linear code over GF(4) with n=44, k=7 and d=20.
This code was found by Heurico 1.16 in 0.158 seconds.