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

generates a code of length 44 over Z4[X]/(X^3+2,2X) who´s minimum homogenous weight is 40.

Homogenous weight enumerator: w(x)=1x^0+51x^40+48x^42+824x^44+48x^46+51x^48+1x^88

The gray image is a code over GF(2) with n=352, k=10 and d=160.
This code was found by Heurico 1.16 in 0.047 seconds.