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

generates a code of length 82 over Z5[X]/(X^2) who´s minimum homogenous weight is 290.

Homogenous weight enumerator: w(x)=1x^0+88x^290+456x^295+800x^300+1032x^305+1392x^310+3108x^315+8904x^320+18012x^325+25048x^330+14356x^335+1268x^340+1176x^345+920x^350+692x^355+484x^360+236x^365+124x^370+12x^375+8x^380+8x^385

The gray image is a linear code over GF(5) with n=410, k=7 and d=290.
This code was found by Heurico 1.16 in 34.4 seconds.