The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  1  1  X  X  1
 0  X  2 3X+2  0 3X+2  2 3X 3X+2  0  2 3X  0 3X+2  2 3X 2X X+2 2X+2 3X  0 3X+2  2 3X  2 3X+2  0 3X 2X X+2 2X+2  X  0 3X+2 2X X+2 2X X+2  2 2X+2 2X+2 3X+2 2X  0
 0  0 2X  0  0  0 2X 2X 2X  0  0 2X 2X  0 2X 2X 2X 2X 2X  0 2X  0  0  0  0 2X 2X  0  0 2X  0 2X  0 2X 2X 2X 2X  0 2X  0 2X  0  0  0
 0  0  0 2X  0  0  0  0  0 2X 2X 2X 2X 2X 2X 2X  0  0 2X 2X 2X  0  0  0  0 2X  0 2X 2X 2X 2X  0 2X 2X  0  0 2X 2X  0  0  0  0  0  0
 0  0  0  0 2X 2X 2X 2X  0 2X  0 2X  0 2X 2X  0  0 2X  0 2X 2X  0  0 2X 2X  0 2X  0  0 2X 2X  0  0 2X 2X  0 2X  0  0  0  0 2X 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+7x^40+72x^41+50x^42+172x^43+475x^44+148x^45+24x^46+20x^47+12x^48+36x^49+5x^50+1x^52+1x^82

The gray image is a code over GF(2) with n=352, k=10 and d=160.
This code was found by Heurico 1.16 in 0.078 seconds.