The generator matrix

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

generates a code of length 44 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 40.

Homogenous weight enumerator: w(x)=1x^0+335x^40+96x^41+808x^42+480x^43+870x^44+288x^45+704x^46+160x^47+311x^48+24x^50+8x^52+9x^56+2x^60

The gray image is a code over GF(2) with n=352, k=12 and d=160.
This code was found by Heurico 1.16 in 3.69 seconds.