The generator matrix

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

generates a code of length 87 over Z7[X]/(X^2) who´s minimum homogenous weight is 506.

Homogenous weight enumerator: w(x)=1x^0+3108x^506+3948x^507+2268x^508+126x^509+1050x^510+1266x^511+546x^512+9618x^513+9576x^514+3696x^515+798x^516+2730x^517+1878x^518+546x^519+11550x^520+9576x^521+4536x^522+378x^523+1974x^524+1764x^525+504x^526+9702x^527+8946x^528+3696x^529+756x^530+2478x^531+1560x^532+462x^533+9240x^534+7056x^535+2268x^536+18x^539+24x^546+6x^553

The gray image is a linear code over GF(7) with n=609, k=6 and d=506.
This code was found by Heurico 1.16 in 6.41 seconds.