The generator matrix

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

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+292x^42+152x^43+244x^44+80x^45+167x^46+24x^47+58x^48+4x^50+1x^62+1x^64

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 1.06 seconds.