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

generates a code of length 79 over Z7[X]/(X^2) who´s minimum homogenous weight is 471.

Homogenous weight enumerator: w(x)=1x^0+1008x^471+252x^473+294x^474+36x^476+756x^478+42x^480+6x^504+6x^511

The gray image is a linear code over GF(7) with n=553, k=4 and d=471.
This code was found by Heurico 1.16 in 39.5 seconds.