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

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+72x^41+73x^42+140x^43+465x^44+138x^45+62x^46+56x^47+6x^48+10x^49+1x^86

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 0.046 seconds.