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

generates a code of length 44 over Z4[X]/(X^2+2) who´s minimum homogenous weight is 40.

Homogenous weight enumerator: w(x)=1x^0+330x^40+128x^41+384x^43+384x^44+384x^45+128x^47+294x^48+14x^56+1x^64

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 35.4 seconds.