The generator matrix

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

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+179x^40+32x^41+210x^42+480x^43+266x^44+480x^45+204x^46+32x^47+144x^48+2x^50+13x^52+4x^56+1x^76

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 0.157 seconds.