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

generates a code of length 78 over Z7[X]/(X^2) who´s minimum homogenous weight is 441.

Homogenous weight enumerator: w(x)=1x^0+114x^441+798x^447+1698x^448+1596x^453+4956x^454+5472x^455+4032x^460+7896x^461+7956x^462+11844x^467+12684x^468+16086x^469+11340x^474+13566x^475+11802x^476+3318x^482+2058x^483+114x^490+96x^497+96x^504+60x^511+24x^518+30x^525+12x^532

The gray image is a linear code over GF(7) with n=546, k=6 and d=441.
This code was found by Heurico 1.16 in 6.9 seconds.