The generator matrix

 1  1  1  1  1  1  1  1  1  1
 0  X  0  0  X  0  X  X a^2*X a^2*X
 0  0  X  0 a^2*X  X a^2*X  0 a^2*X a^2*X
 0  0  0  X  X  X a*X a^2*X  0  X

generates a code of length 10 over F4[X]/(X^2) who´s minimum homogenous weight is 24.

Homogenous weight enumerator: w(x)=1x^0+78x^24+48x^28+768x^30+63x^32+48x^36+18x^40

The gray image is a linear code over GF(4) with n=40, k=5 and d=24.
This code was found by Heurico 1.16 in -3.24e-008 seconds.