The generator matrix

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

generates a code of length 14 over F4[X]/(X^2) who´s minimum homogenous weight is 28.

Homogenous weight enumerator: w(x)=1x^0+168x^28+387x^32+1284x^36+5358x^40+8064x^44+756x^48+276x^52+90x^56

The gray image is a linear code over GF(4) with n=56, k=7 and d=28.
This code was found by Heurico 1.16 in 0.291 seconds.