The generator matrix

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

generates a code of length 86 over Z7[X]/(X^2) who´s minimum homogenous weight is 508.

Homogenous weight enumerator: w(x)=1x^0+3906x^508+2646x^509+168x^511+3402x^515+882x^516+108x^518+588x^522+2646x^523+42x^525+2394x^529+6x^532+18x^553

The gray image is a linear code over GF(7) with n=602, k=5 and d=508.
This code was found by Heurico 1.16 in 7.51 seconds.