The generator matrix

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

generates a code of length 56 over F4[X]/(X^2) who´s minimum homogenous weight is 157.

Homogenous weight enumerator: w(x)=1x^0+480x^157+768x^158+360x^159+51x^160+1080x^161+1260x^162+456x^163+60x^164+1704x^165+1608x^166+588x^167+45x^168+1464x^169+1440x^170+336x^171+60x^172+1080x^173+936x^174+336x^175+6x^176+816x^177+660x^178+168x^179+15x^180+288x^181+240x^182+60x^183+9x^184+6x^188+3x^196

The gray image is a linear code over GF(4) with n=224, k=7 and d=157.
This code was found by Heurico 1.16 in 91.3 seconds.