The generator matrix

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

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+1x^44+22x^45+2x^46+6x^47

The gray image is a code over GF(2) with n=352, k=5 and d=176.
This code was found by Heurico 1.16 in 0.031 seconds.