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

generates a code of length 67 over Z7[X]/(X^2) who´s minimum homogenous weight is 378.

Homogenous weight enumerator: w(x)=1x^0+276x^378+42x^381+546x^382+1260x^384+1188x^385+420x^386+1176x^388+4410x^389+4284x^391+2670x^392+1680x^393+2394x^395+8190x^396+6048x^398+4230x^399+5670x^400+5922x^402+17598x^403+11172x^405+5064x^406+6636x^407+4872x^409+12474x^410+6048x^412+2922x^413+120x^420+108x^427+90x^434+60x^441+30x^448+24x^455+18x^462+6x^469

The gray image is a linear code over GF(7) with n=469, k=6 and d=378.
This code was found by Heurico 1.16 in 5.79 seconds.