The generator matrix

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

generates a code of length 44 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 39.

Homogenous weight enumerator: w(x)=1x^0+188x^39+730x^40+1470x^41+2299x^42+2170x^43+3025x^44+2080x^45+2177x^46+1134x^47+621x^48+330x^49+73x^50+42x^51+29x^52+8x^53+3x^54+2x^55+2x^56

The gray image is a code over GF(2) with n=352, k=14 and d=156.
This code was found by Heurico 1.16 in 1.75 seconds.