MinT - Best Known (250−159, 250, s)-Nets in Base 3

Best Known (250−159, 250, s)-Nets in Base 3

Digital (91, 250, 64)-net over F₃, using

t-expansion [i] based on digital (89, 250, 64)-net over F₃, using
- net from sequence [i] based on digital (89, 63)-sequence over F₃, using
  - base reduction for sequences [i] 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 (91, 250, 96)-net over F₃, using

There is no digital (91, 250, 404)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²⁵⁰, 404, F₃, 159) (dual of [404, 154, 160]-code), but
- residual code [i] would yield linear OA(3⁹¹, 244, F₃, 53) (dual of [244, 153, 54]-code), but
  - 1 times truncation [i] would yield linear OA(3⁹⁰, 243, F₃, 52) (dual of [243, 153, 53]-code), but
    - the Johnson bound shows that NÂ â‰¤ 9 342763 491628 254791 739936 787780 423565 169992 498824 859636 256969 626845 605352 < 3¹⁵³ [i]

There is no (91, 250, 409)-net in base 3, because

1 times m-reduction [i] would yield (91, 249, 409)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 68239 483198 878114 973269 276029 637600 365704 458554 703828 724726 259867 497797 790491 168448 486538 907837 943458 590027 352976 073163 > 3²⁴⁹ [i]