The generator matrix

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

generates a code of length 44 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 42.

Homogenous weight enumerator: w(x)=1x^0+46x^42+50x^43+286x^44+108x^45+16x^46+2x^47+1x^52+1x^54+1x^66

The gray image is a code over GF(2) with n=352, k=9 and d=168.
This code was found by Heurico 1.16 in 0.032 seconds.