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

generates a code of length 79 over Z7[X]/(X^2) who´s minimum homogenous weight is 463.

Homogenous weight enumerator: w(x)=1x^0+1386x^463+882x^464+1344x^465+204x^469+3066x^470+1722x^471+1176x^472+36x^476+1470x^477+378x^478+672x^479+54x^483+2310x^484+1134x^485+924x^486+24x^490+24x^497

The gray image is a linear code over GF(7) with n=553, k=5 and d=463.
This code was found by Heurico 1.16 in 34.7 seconds.