MinT - Best Known (249−161, 249, s)-Nets in Base 3

Best Known (249−161, 249, s)-Nets in Base 3

Digital (88, 249, 63)-net over F₃, using

net from sequence [i] based on digital (88, 62)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 62)-sequence over F₉, using
  - s-reduction based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

Digital (88, 249, 84)-net over F₃, using

There is no digital (88, 249, 376)-net over F₃, because

2 times m-reduction [i] would yield digital (88, 247, 376)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²⁴⁷, 376, F₃, 159) (dual of [376, 129, 160]-code), but
  - residual code [i] would yield linear OA(3⁸⁸, 216, F₃, 53) (dual of [216, 128, 54]-code), but
    - 1 times truncation [i] would yield linear OA(3⁸⁷, 215, F₃, 52) (dual of [215, 128, 53]-code), but
      - the Johnson bound shows that NÂ â‰¤ 11 307865 685880 246142 774871 856086 316274 707673 744713 882597 679605 < 3¹²⁸ [i]

There is no (88, 249, 387)-net in base 3, because

1 times m-reduction [i] would yield (88, 248, 387)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 22122 788516 257461 421368 379533 800556 579928 382364 638948 283174 962166 689599 635961 616154 861890 252545 641461 364520 898368 397409 > 3²⁴⁸ [i]