The generator matrix

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

generates a code of length 44 over Z4[X]/(X^2+2X+2) who´s minimum homogenous weight is 43.

Homogenous weight enumerator: w(x)=1x^0+28x^43+82x^44+8x^46+4x^48+4x^51+1x^56

The gray image is a code over GF(2) with n=352, k=7 and d=172.
This code was found by Heurico 1.16 in 0.031 seconds.