The generator matrix

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

generates a code of length 54 over Z7[X]/(X^2) who´s minimum homogenous weight is 310.

Homogenous weight enumerator: w(x)=1x^0+2898x^310+5502x^311+1050x^312+252x^314+24x^315+12474x^317+16548x^318+2730x^319+672x^321+168x^322+15750x^324+16254x^325+1974x^326+1134x^328+144x^329+18270x^331+19320x^332+2478x^333+6x^350

The gray image is a linear code over GF(7) with n=378, k=6 and d=310.
This code was found by Heurico 1.16 in 3.71 seconds.