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

generates a code of length 83 over Z7[X]/(X^2) who´s minimum homogenous weight is 489.

Homogenous weight enumerator: w(x)=1x^0+1218x^489+2982x^490+1764x^491+1470x^496+2934x^497+588x^498+462x^503+1110x^504+1764x^505+966x^510+1524x^511+6x^525+12x^532+6x^539

The gray image is a linear code over GF(7) with n=581, k=5 and d=489.
This code was found by Heurico 1.16 in 0.23 seconds.