The generator matrix

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

generates a code of length 15 over F4[X]/(X^2) who´s minimum homogenous weight is 28.

Homogenous weight enumerator: w(x)=1x^0+93x^28+246x^32+360x^36+579x^40+852x^44+12288x^45+945x^48+636x^52+309x^56+75x^60

The gray image is a linear code over GF(4) with n=60, k=7 and d=28.
This code was found by Heurico 1.16 in 0.292 seconds.