The generator matrix

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

generates a code of length 78 over Z7[X]/(X^2) who´s minimum homogenous weight is 458.

Homogenous weight enumerator: w(x)=1x^0+3150x^458+672x^459+882x^461+144x^462+3654x^465+588x^466+294x^468+138x^469+2646x^472+336x^473+882x^475+6x^476+2898x^479+462x^480+24x^483+12x^490+18x^497

The gray image is a linear code over GF(7) with n=546, k=5 and d=458.
This code was found by Heurico 1.16 in 15.4 seconds.