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

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+63x^40+896x^44+63x^48+1x^88

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 0.062 seconds.