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

generates a code of length 81 over Z5[X]/(X^2) who´s minimum homogenous weight is 290.

Homogenous weight enumerator: w(x)=1x^0+288x^290+652x^295+1100x^300+20x^304+1288x^305+400x^309+1384x^310+3200x^314+1364x^315+12800x^319+1288x^320+25600x^324+1556x^325+20480x^329+1464x^330+1316x^335+1300x^340+956x^345+788x^350+424x^355+300x^360+128x^365+24x^370+4x^380

The gray image is a linear code over GF(5) with n=405, k=7 and d=290.
This code was found by Heurico 1.16 in 34.2 seconds.