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  1  2  1  1  1  1  1  1  0  0 2X+2  1  1
 0  X 2X+2 X+2  0 X+2 2X+2 3X X+2  0 3X 2X+2 2X 3X+2  2 3X  0 X+2 2X+2 3X  0 X+2  0 3X+2 2X+2  2 3X 3X 2X X+2  2  X 2X+2  0 2X X+2 3X+2 2X+2 2X+2  2  X  X 2X+2  2
 0  0 2X  0  0  0 2X  0 2X  0 2X  0  0 2X  0 2X  0  0 2X 2X 2X  0 2X  0 2X  0  0 2X 2X 2X 2X 2X  0  0 2X  0 2X  0  0 2X  0  0  0 2X
 0  0  0 2X  0  0  0  0  0  0  0 2X  0 2X 2X  0 2X 2X 2X 2X 2X 2X  0 2X 2X 2X  0 2X  0 2X 2X  0  0 2X 2X  0  0  0  0  0  0  0 2X 2X
 0  0  0  0 2X  0 2X 2X 2X  0  0  0 2X 2X 2X 2X 2X  0 2X  0  0 2X  0 2X 2X  0 2X 2X 2X  0  0  0  0 2X 2X  0  0  0 2X 2X  0 2X 2X  0
 0  0  0  0  0 2X  0 2X 2X 2X 2X 2X 2X  0 2X  0  0 2X  0 2X  0 2X 2X  0 2X  0  0 2X 2X  0 2X  0  0 2X 2X  0  0  0  0  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+204x^40+296x^42+256x^43+574x^44+256x^45+240x^46+202x^48+8x^50+10x^52+1x^80

The gray image is a code over GF(2) with n=352, k=11 and d=160.
This code was found by Heurico 1.16 in 0.11 seconds.