The generator matrix

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

generates a code of length 44 over Z4[X]/(X^2+2) who´s minimum homogenous weight is 38.

Homogenous weight enumerator: w(x)=1x^0+244x^38+1314x^39+3225x^40+4732x^41+7547x^42+9946x^43+11673x^44+9826x^45+7816x^46+4846x^47+2616x^48+1072x^49+475x^50+130x^51+59x^52+6x^54+4x^55+2x^56+2x^61

The gray image is a code over GF(2) with n=352, k=16 and d=152.
This code was found by Heurico 1.16 in 22.4 seconds.