The generator matrix

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

generates a code of length 84 over Z7[X]/(X^2) who´s minimum homogenous weight is 501.

Homogenous weight enumerator: w(x)=1x^0+1176x^501+294x^504+882x^508+42x^511+6x^539

The gray image is a linear code over GF(7) with n=588, k=4 and d=501.
This code was found by Heurico 1.16 in 0.0848 seconds.