The generator matrix

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

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

Homogenous weight enumerator: w(x)=1x^0+118x^44+4x^46+4x^50+1x^56

The gray image is a code over GF(2) with n=352, k=7 and d=176.
This code was found by Heurico 1.16 in 0.031 seconds.