The generator matrix

 1  0  0  1  1  1  1  1  1  1  1  1  X  1  1  1  1  1  1  1 6X  1  1  1  1  1  1 6X 5X  1 3X  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 4X  1  1  1  1  1 5X  1  1  1  1 2X  1  1  1  1  1  1  1  1  1  1  1  1  1  X
 0  1  0 5X 3X 6X  1 5X+3  2 5X+1 4X+1 6X+1  1  3 X+6  4 5X+2 3X+6 4X+3 4X+2  1 5X+4 2X+4 3X+3 3X+2 4X+6  6  1  1 6X+6  1 3X+4 2X  1 6X+2 3X+5 4X+4 X+4 5X+6 X+2 X+1 5X+3 3X+3  X 2X+1 6X+5 2X+5 X+5 3X 4X+5 X+1  5 5X+4 2X+4 2X+6  1 6X+6 3X+3  1 2X+3 3X+2 2X+2 2X+3 4X+2  1  0 2X+6 X+2 4X+4  1  3 2X+5  4 5X+2  4 2X+6  3  5 3X+2 5X  6 6X 5X  1
 0  0  1 5X+1  3 5X+2  2 6X+2 4X+2 5X+5  6 5X+3 3X+3 X+5 3X 6X+1 6X+5 6X+6 3X+1 2X 5X+6 3X+6 6X+4  4 2X+6 X+3 X+4 2X+4  5 4X+5 6X+1 2X+3 X+6 2X+4 3X+4  3  2 5X 2X+1 5X+3 4X 6X  6 3X+5 5X+1  1 3X+2 2X+5 4X+4 X+6 4X+3 3X+4 3X+5 5X+6 4X+2 3X+2  0 3X+3 6X+1 2X+2  4 3X+2 2X 5X 3X+6 6X+5 5X+6 2X+1 5X+5  X 5X+4 3X+3  1 6X+4 4X+4 4X+5  X 3X+1 3X 6X+3 6X+2 X+2 6X+1 5X

generates a code of length 84 over Z7[X]/(X^2) who´s minimum homogenous weight is 488.

Homogenous weight enumerator: w(x)=1x^0+2940x^488+3444x^489+1590x^490+840x^491+1512x^492+2016x^493+546x^494+8274x^495+9072x^496+3216x^497+2142x^498+2982x^499+2940x^500+546x^501+8820x^502+10122x^503+3300x^504+2604x^505+2730x^506+2772x^507+504x^508+9282x^509+7560x^510+2778x^511+2646x^512+3066x^513+2562x^514+462x^515+7728x^516+6846x^517+1752x^518+30x^525+24x^532

The gray image is a linear code over GF(7) with n=588, k=6 and d=488.
This code was found by Heurico 1.16 in 6.18 seconds.