The generator matrix

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

generates a code of length 13 over Z4[X]/(X^2+2,2X) who´s minimum homogenous weight is 4.

Homogenous weight enumerator: w(x)=1x^0+53x^4+48x^6+340x^8+424x^10+512x^11+1266x^12+3072x^13+936x^14+512x^15+721x^16+120x^18+169x^20+8x^22+10x^24

The gray image is a code over GF(2) with n=52, k=13 and d=8.
This code was found by Heurico 1.16 in 0.0742 seconds.