The generator matrix

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

generates a code of length 11 over F4[X]/(X^2) who´s minimum homogenous weight is 24.

Homogenous weight enumerator: w(x)=1x^0+165x^24+165x^28+165x^32+3072x^33+330x^36+165x^40+33x^44

The gray image is a linear code over GF(4) with n=44, k=6 and d=24.
This code was found by Heurico 1.16 in 0.0138 seconds.