The generator matrix

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

generates a code of length 44 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 42.

Homogenous weight enumerator: w(x)=1x^0+9x^42+34x^43+173x^44+26x^45+6x^46+2x^47+2x^48+1x^54+2x^61

The gray image is a code over GF(2) with n=352, k=8 and d=168.
This code was found by Heurico 1.16 in 0.031 seconds.