The generator matrix

 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  1  1  1  1  1  1  1  1  1  1  1  1  1  X  1  X  1  X  1  X  1  X 2X+2 2X+2
 0  2  0  0  0 2X+2  2 2X+2  0 2X 2X  0 2X+2 2X+2  2  2  0 2X  0  0 2X+2 2X+2 2X+2 2X+2 2X 2X+2 2X 2X+2  0 2X  2 2X 2X+2 2X+2 2X  0 2X  2  0 2X+2  0 2X+2 2X+2  2
 0  0  2  0 2X+2 2X+2 2X+2 2X  2 2X  0 2X+2 2X+2 2X+2  0  0  0 2X  2  2  2  0 2X  2  2 2X 2X+2  0 2X+2  2 2X+2 2X+2 2X+2  2 2X 2X+2  0  2  0  2 2X 2X+2 2X+2 2X+2
 0  0  0  2 2X+2 2X  2  2  0  0 2X+2 2X+2  0  2 2X+2 2X  0  2 2X+2 2X 2X+2  0 2X+2 2X  0  0 2X 2X  0 2X+2 2X+2  2  2 2X 2X 2X  0  2  2  0  2  0 2X 2X+2
 0  0  0  0 2X 2X 2X  0 2X 2X 2X  0  0  0 2X 2X 2X 2X 2X  0  0  0 2X 2X 2X 2X  0  0  0  0  0  0 2X 2X  0  0 2X  0 2X  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+223x^40+192x^42+256x^43+764x^44+256x^45+192x^46+109x^48+44x^52+10x^56+1x^72

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 12.9 seconds.