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

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+102x^448+420x^451+336x^454+1194x^455+630x^456+1008x^457+2688x^458+3024x^461+3972x^462+2100x^463+2268x^464+4578x^465+5376x^468+6426x^469+3150x^470+3318x^471+6510x^472+11760x^475+11712x^476+5460x^477+5166x^478+9786x^479+8316x^482+7272x^483+3066x^484+2646x^485+4830x^486+132x^490+120x^497+72x^504+78x^511+54x^518+36x^525+24x^532+12x^539+6x^546

The gray image is a linear code over GF(7) with n=553, k=6 and d=448.
This code was found by Heurico 1.16 in 7.04 seconds.