The generator matrix

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

generates a code of length 54 over Z7[X]/(X^2) who´s minimum homogenous weight is 287.

Homogenous weight enumerator: w(x)=1x^0+234x^287+996x^294+1488x^301+1890x^308+1992x^315+14406x^318+2118x^322+86436x^325+2340x^329+2160x^336+1794x^343+1056x^350+498x^357+198x^364+42x^371

The gray image is a linear code over GF(7) with n=378, k=6 and d=287.
This code was found by Heurico 1.16 in 11.5 seconds.