The generator matrix

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

generates a code of length 11 over F4[X]/(X^2) who´s minimum homogenous weight is 28.

Homogenous weight enumerator: w(x)=1x^0+111x^28+96x^30+267x^32+288x^34+246x^36+12x^40+3x^44

The gray image is a linear code over GF(4) with n=44, k=5 and d=28.
This code was found by Heurico 1.16 in 0 seconds.