The generator matrix

 1  0  1  1  1 3X+2  1  1  2  1  1 3X  1  1  0  1  1 3X+2  1  1  2  1  1 3X  1  1  0  1  1 3X+2 3X  1  1  1  1  0  X  1  1 2X  1  2  1  1
 0  1 X+1 3X+2 2X+3  1 X+3  2  1 3X 2X+1  1  0 X+1  1 3X+2 2X+3  1  2 X+3  1 3X 2X+1  1  0 X+1  1 X+3 3X+2  1  1 2X+3  0 3X+2 2X+3  1 3X+2  3 2X  1  2  1 2X+1  0
 0  0 2X  0  0  0  0 2X 2X 2X 2X 2X  0  0  0  0  0 2X 2X 2X 2X 2X 2X  0  0  0  0 2X 2X 2X  0  0 2X  0 2X 2X  0 2X 2X 2X 2X 2X 2X 2X
 0  0  0 2X  0 2X 2X 2X 2X  0 2X  0 2X  0 2X  0 2X 2X  0 2X  0 2X  0  0  0  0 2X  0  0 2X 2X  0  0  0  0 2X 2X 2X 2X  0  0 2X  0 2X
 0  0  0  0 2X  0 2X 2X 2X 2X  0 2X  0 2X  0  0 2X 2X 2X  0 2X 2X  0  0 2X  0 2X 2X  0  0 2X  0  0 2X 2X  0  0 2X  0  0  0  0 2X 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+72x^40+200x^41+227x^42+392x^43+304x^44+360x^45+216x^46+184x^47+69x^48+16x^49+2x^50+2x^56+3x^58

The gray image is a code over GF(2) with n=352, k=11 and d=160.
This code was found by Heurico 1.16 in 0.063 seconds.