The generator matrix

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

generates a code of length 97 over Z7[X]/(X^2) who´s minimum homogenous weight is 560.

Homogenous weight enumerator: w(x)=1x^0+768x^560+756x^561+378x^563+630x^564+2856x^566+3834x^567+3654x^568+1806x^570+2142x^571+5082x^573+5250x^574+5670x^575+2898x^577+3024x^578+6426x^580+7608x^581+7014x^582+5796x^584+5586x^585+9870x^587+9180x^588+8190x^589+3528x^591+3024x^592+4578x^594+4158x^595+3528x^596+102x^602+72x^609+90x^616+30x^623+42x^630+24x^637+12x^644+24x^651+6x^658+12x^665

The gray image is a linear code over GF(7) with n=679, k=6 and d=560.
This code was found by Heurico 1.16 in 45.8 seconds.