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

generates a code of length 53 over Z7[X]/(X^2) who´s minimum homogenous weight is 280.

Homogenous weight enumerator: w(x)=1x^0+168x^280+948x^287+1350x^294+2802x^301+11130x^308+38358x^315+56766x^322+2244x^329+1962x^336+1098x^343+600x^350+204x^357+18x^364

The gray image is a linear code over GF(7) with n=371, k=6 and d=280.
This code was found by Heurico 1.16 in 11.3 seconds.