The generator matrix

 1  1  1  1  X  1  1  1  1  1  1
 0  X  0  0  X  0  X X+2  2  0  0
 0  0  X  0  X  X  X X+2  X  0  0
 0  0  0  X  X  X  0 X+2  0  0  0
 0  0  0  0  2  0  0  0  2  0  0
 0  0  0  0  0  2  0  0  2  0  0
 0  0  0  0  0  0  2  0  2  0  0
 0  0  0  0  0  0  0  2  2  0  0

generates a code of length 11 over Z4[X]/(X^2+2,2X) who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+68x^4+264x^6+710x^8+2008x^10+2048x^11+2092x^12+632x^14+321x^16+40x^18+8x^20

The gray image is a code over GF(2) with n=44, k=13 and d=8.
This code was found by Heurico 1.16 in 0.0614 seconds.