The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+380x^42+260x^44+380x^46+1x^56+2x^60

The gray image is a code over GF(2) with n=352, k=10 and d=168.
This code was found by Heurico 1.16 in 0.078 seconds.