The generator matrix

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

generates a code of length 79 over Z7[X]/(X^2) who´s minimum homogenous weight is 448.

Homogenous weight enumerator: w(x)=1x^0+84x^448+474x^455+426x^462+2058x^468+414x^469+12348x^475+282x^476+186x^483+120x^490+150x^497+96x^504+24x^511+54x^518+48x^525+18x^532+18x^539+6x^546

The gray image is a linear code over GF(7) with n=553, k=5 and d=448.
This code was found by Heurico 1.16 in 0.447 seconds.