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  X  X
 0  X  0 X^2+X X^2 X^2+X+2 X^2+2  X  0 X^2+X X^2+2 X+2  2  X X^2+2 X^2+X+2  X  0 X^2+2 X^2+X X^2+X  2 X^2 X+2 X+2  0 X^2+X X^2+X  0 X^2+2 X^2+2 X+2 X+2  2  2  2 X^2+X X^2+2 X^2+X+2 X^2+X+2 X^2 X^2 X^2+X X^2+X+2
 0  0 X^2+2  0 X^2+2 X^2  0 X^2  2  2  2  2 X^2 X^2+2 X^2 X^2+2 X^2  0  0  0 X^2+2 X^2 X^2  2 X^2+2  2 X^2  2 X^2+2 X^2+2  2  0 X^2+2  2 X^2+2  2 X^2+2 X^2  2  0  2  0  0 X^2
 0  0  0  2  2  0  2  2  0  2  2  0  0  2  2  0  0  2  0  0  2  2  0  2  0  2  2  0  2  0  0  2  2  0  2  2  0  0  0  2  0  2  2  0

generates a code of length 44 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 41.

Homogenous weight enumerator: w(x)=1x^0+88x^41+44x^42+172x^43+421x^44+164x^45+44x^46+84x^47+1x^48+4x^49+1x^84

The gray image is a code over GF(2) with n=352, k=10 and d=164.
This code was found by Heurico 1.16 in 0.609 seconds.