MinT - Best Known (13, 13+73, s)-Nets in Base 8

Best Known (13, 13+73, s)-Nets in Base 8

Digital (13, 86, 48)-net over F₈, using

t-expansion [i] based on digital (11, 86, 48)-net over F₈, using
- net from sequence [i] based on digital (11, 47)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 11 and N(F) ≥ 48, using
    - a function field by SÃ©mirat [i]

Digital (13, 86, 56)-net over F₈, using

There is no digital (13, 86, 236)-net over F₈, because

1 times m-reduction [i] would yield digital (13, 85, 236)-net over F₈, but
- extracting embedded orthogonal array [i] would yield linear OA(8⁸⁵, 236, F₈, 72) (dual of [236, 151, 73]-code), but
  - residual code [i] would yield OA(8¹³, 163, S₈, 9), but
    - 1 times truncation [i] would yield OA(8¹², 162, S₈, 8), but
      - the linear programming bound shows that MÂ â‰¥ 12 319920 743776 256000 / 177 696713 > 8¹² [i]

There is no (13, 86, 242)-net in base 8, because

22 times m-reduction [i] would yield (13, 64, 242)-net in base 8, but
- extracting embedded orthogonal array [i] would yield OA(8⁶⁴, 242, S₈, 51), but
  - the linear programming bound shows that MÂ â‰¥ 70 232740 660466 146387 371112 364270 550714 612400 798807 017251 810317 649288 784222 788532 961057 620751 780826 891846 178327 052743 502456 032729 337117 589164 303319 040000 / 10982 962511 522080 424097 485508 217442 568541 585265 006918 081487 297620 694440 959423 507493 470379 310181 > 8⁶⁴ [i]