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

generates a code of length 84 over Z7[X]/(X^2) who´s minimum homogenous weight is 483.

Homogenous weight enumerator: w(x)=1x^0+384x^483+42x^486+492x^490+756x^493+426x^497+4536x^500+342x^504+9072x^507+228x^511+120x^518+90x^525+66x^532+84x^539+48x^546+78x^553+12x^560+24x^567+6x^581

The gray image is a linear code over GF(7) with n=588, k=5 and d=483.
This code was found by Heurico 1.16 in 0.486 seconds.