The generator matrix

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

generates a code of length 34 over Z4[X]/(X^2+2,2X) who´s minimum homogenous weight is 28.

Homogenous weight enumerator: w(x)=1x^0+480x^28+1534x^32+1664x^36+384x^40+32x^44+1x^64

The gray image is a code over GF(2) with n=136, k=12 and d=56.
This code was found by Heurico 1.16 in 71.5 seconds.