The generator matrix

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

generates a code of length 44 over Z4[X]/(X^2+2) who´s minimum homogenous weight is 40.

Homogenous weight enumerator: w(x)=1x^0+132x^40+428x^41+599x^42+754x^43+584x^44+600x^45+382x^46+296x^47+183x^48+72x^49+25x^50+22x^51+12x^52+4x^53+1x^54+1x^62

The gray image is a code over GF(2) with n=352, k=12 and d=160.
This code was found by Heurico 1.16 in 0.172 seconds.