The generator matrix

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

generates a code of length 20 over Z4[X]/(X^2+2,2X) who´s minimum homogenous weight is 16.

Homogenous weight enumerator: w(x)=1x^0+302x^16+192x^18+1072x^20+192x^22+272x^24+16x^28+1x^32

The gray image is a code over GF(2) with n=80, k=11 and d=32.
This code was found by Heurico 1.16 in 8.33 seconds.