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  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  X  1  1  X  1  1  X  X  X  X  0  1  X  1
 0  X  0  0  X  X  0  X 2X 4X 3X 4X  0 2X 4X 4X  X 2X 2X  X 3X 3X  X 3X 4X 3X 2X 2X 4X  X 4X 4X 2X 2X 3X 2X 4X  X 3X  X 2X 4X 2X  X 4X 4X 3X 3X 3X 3X 2X 4X  X 2X  X 4X  X 2X 3X 3X 4X  X 3X 2X 2X 3X 4X  0  X 3X  0  X 3X  X  X 2X 2X  X  X 2X  0 4X
 0  0  X  0 3X 2X  X 4X  0  X  X  X 3X 2X  0 2X 3X  X 2X 4X  0 3X 2X 3X  0  X 4X 3X 4X 3X  X 2X 3X  X  0 2X 4X 4X 3X 2X 3X 2X 4X 2X  X 4X  X 2X  0 4X 4X  0 4X  0  X 2X  X 4X 2X 4X 3X  X 4X  0 2X  X 4X 2X 3X  X 2X 3X 4X  X 4X 4X 3X 2X  0 3X  X 2X
 0  0  0  X 3X  X 4X 3X 3X 3X  0  X  X  0 3X  X 2X 2X 3X  0 4X  X 3X 3X  X  X 3X 2X 2X  0 2X 2X  0  0  0 2X  X 2X 4X 4X  X 3X  0 2X 4X 4X 2X  X 3X  0 2X 4X 4X  X 3X 4X  0  X 3X  X 3X 4X 4X 4X 4X 3X  0  X  X 4X 2X 4X 2X 2X 4X 2X  0  0 2X 3X  X  X

generates a code of length 82 over Z5[X]/(X^2) who´s minimum homogenous weight is 316.

Homogenous weight enumerator: w(x)=1x^0+140x^316+252x^320+460x^321+116x^325+580x^326+176x^330+900x^331+420x^336+36x^345+8x^350+8x^355+12x^370+16x^380

The gray image is a linear code over GF(5) with n=410, k=5 and d=316.
This code was found by Heurico 1.16 in 67.3 seconds.