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

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+459x^40+96x^41+708x^42+440x^43+890x^44+376x^45+564x^46+104x^47+314x^48+8x^49+100x^50+30x^52+4x^54+1x^56+1x^64

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 23.4 seconds.