The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+46x^40+248x^41+220x^42+328x^43+388x^44+328x^45+184x^46+248x^47+41x^48+12x^50+2x^52+2x^68

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 0.078 seconds.