The generator matrix

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

generates a code of length 14 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 10.

Homogenous weight enumerator: w(x)=1x^0+274x^10+1696x^11+5690x^12+14818x^13+20380x^14+15224x^15+5583x^16+1516x^17+322x^18+24x^19+6x^20+2x^21

The gray image is a code over GF(2) with n=112, k=16 and d=40.
This code was found by Heurico 1.16 in 4.81 seconds.