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

generates a code of length 56 over F4[X]/(X^2) who´s minimum homogenous weight is 144.

Homogenous weight enumerator: w(x)=1x^0+66x^144+216x^148+393x^152+432x^156+387x^160+768x^162+381x^164+4608x^166+423x^168+6912x^170+384x^172+378x^176+450x^180+216x^184+180x^188+96x^192+57x^196+21x^200+12x^204+3x^216

The gray image is a linear code over GF(4) with n=224, k=7 and d=144.
This code was found by Heurico 1.16 in 2.4 seconds.