MinT - Best Known (249−133, 249, s)-Nets in Base 3

Best Known (249−133, 249, s)-Nets in Base 3

(249−133, 249, 74)-Net over F₃ — Constructive and digital

Digital (116, 249, 74)-net over F₃, using

t-expansion [i] based on digital (107, 249, 74)-net over F₃, using
- net from sequence [i] based on digital (107, 73)-sequence over F₃, using
  - base reduction for sequences [i] based on digital (17, 73)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
      - a function field by GarcÃa and Quoos [i]

(249−133, 249, 120)-Net over F₃ — Digital

Digital (116, 249, 120)-net over F₃, using

t-expansion [i] based on digital (113, 249, 120)-net over F₃, using
- net from sequence [i] based on digital (113, 119)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 113 and N(F) ≥ 120, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(249−133, 249, 725)-Net in Base 3 — Upper bound on s

There is no (116, 249, 726)-net in base 3, because

1 times m-reduction [i] would yield (116, 248, 726)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 22454 983939 375897 606279 131010 562992 673516 031033 290461 625996 421996 853376 345906 175625 077981 653141 129887 864704 739684 433741 > 3²⁴⁸ [i]