The generator matrix

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

generates a code of length 44 over Z4[X]/(X^2+2,2X) who´s minimum homogenous weight is 39.

Homogenous weight enumerator: w(x)=1x^0+270x^39+338x^40+458x^41+431x^42+490x^43+387x^44+450x^45+308x^46+346x^47+211x^48+182x^49+71x^50+78x^51+39x^52+30x^53+6x^54

The gray image is a code over GF(2) with n=176, k=12 and d=78.
This code was found by Heurico 1.16 in 67 seconds.