The generator matrix

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

generates a code of length 52 over F4[X]/(X^2) who´s minimum homogenous weight is 138.

Homogenous weight enumerator: w(x)=1x^0+660x^138+924x^139+375x^140+780x^141+3360x^142+3720x^143+1446x^144+1812x^145+7932x^146+8112x^147+3264x^148+3588x^149+15852x^150+15432x^151+5061x^152+6564x^153+24408x^154+21216x^155+6999x^156+7764x^157+27816x^158+22056x^159+6909x^160+6444x^161+21036x^162+15216x^163+3672x^164+3180x^165+8268x^166+4872x^167+750x^168+540x^169+1260x^170+612x^171+114x^172+48x^173+36x^176+18x^180+12x^184+6x^188+6x^192+3x^200

The gray image is a linear code over GF(4) with n=208, k=9 and d=138.
This code was found by Heurico 1.16 in 289 seconds.