The generator matrix

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

generates a code of length 28 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 22.

Homogenous weight enumerator: w(x)=1x^0+97x^22+942x^23+2877x^24+7870x^25+15176x^26+24242x^27+27859x^28+25688x^29+14967x^30+7434x^31+2823x^32+854x^33+188x^34+38x^35+7x^36+4x^37+2x^38+1x^40+2x^42

The gray image is a code over GF(2) with n=224, k=17 and d=88.
This code was found by Heurico 1.16 in 48.6 seconds.