The generator matrix

 1  0  0  1  1  1 2X+2  1  1  1 3X+2  X 3X+2  1  0 3X+2
 0  1  0 2X+2 2X+3 3X+1  1 3X+2 2X+1 3X+1 3X+2  1  2 3X+2 2X+2  1
 0  0  1 3X+3 X+1 2X+2  1 X+1 2X+3  X  1  1  1  0  1 2X+3
 0  0  0 2X 2X 2X 2X  0  0  0 2X  0  0 2X 2X 2X

generates a code of length 16 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 13.

Homogenous weight enumerator: w(x)=1x^0+268x^13+958x^14+1520x^15+2739x^16+1544x^17+860x^18+240x^19+44x^20+12x^21+6x^22

The gray image is a code over GF(2) with n=128, k=13 and d=52.
This code was found by Heurico 1.16 in 5.33 seconds.