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+1)X aX  X  X aX aX (a+1)X aX  0 (a+1)X (a+1)X  X aX aX (a+1)X (a+1)X  X aX  X (a+1)X  X  X  X (a+1)X  0  0 aX aX  X  X (a+1)X  X  X (a+1)X  X (a+1)X  0 aX aX  0  X aX  0
 0  0  X  0  0  0  0  X  X  X aX (a+1)X (a+1)X aX  0 (a+1)X aX  X aX (a+1)X (a+1)X (a+1)X  0 (a+1)X aX  0  X  0  X  X  0  X  X aX aX  X  X  X  0 aX  0  0 (a+1)X (a+1)X  0  X  0 aX (a+1)X aX (a+1)X (a+1)X  0  0  0  0
 0  0  0  X  0  0  X (a+1)X (a+1)X aX  0  0  X  0  X (a+1)X aX  X (a+1)X (a+1)X  X  X  X aX  X  0 (a+1)X aX (a+1)X aX  X (a+1)X  0  X  0  0  0  0  X  0 (a+1)X  X aX aX aX (a+1)X  X aX (a+1)X (a+1)X aX  0  0  X  0  0
 0  0  0  0  X  0 (a+1)X  0 (a+1)X  X aX  X  0  X aX (a+1)X aX aX aX aX  0 aX aX aX (a+1)X aX  0 aX  0 (a+1)X  0  X  0  X aX  X  0  0 aX (a+1)X  X (a+1)X (a+1)X  X aX (a+1)X  0 aX  0 (a+1)X aX (a+1)X (a+1)X aX  X  0
 0  0  0  0  0  X  X  X (a+1)X (a+1)X (a+1)X (a+1)X  0  X (a+1)X  X  X  0 aX  X  X (a+1)X  0  X  0 aX (a+1)X aX (a+1)X (a+1)X (a+1)X aX aX aX  0 (a+1)X  0 aX  0  0 (a+1)X  0 (a+1)X (a+1)X  0 (a+1)X (a+1)X aX  X aX (a+1)X  X  0 (a+1)X  X  0

generates a code of length 56 over F4[X,sigma]/(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 2.42 seconds.