MinT - Best Known (245−126, 245, s)-Nets in Base 3

Best Known (245−126, 245, s)-Nets in Base 3

(245−126, 245, 76)-Net over F₃ — Constructive and digital

Digital (119, 245, 76)-net over F₃, using

net from sequence [i] based on digital (119, 75)-sequence over F₃, using
- base reduction for sequences [i] based on digital (22, 75)-sequence over F₉, using
  - s-reduction based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(245−126, 245, 120)-Net over F₃ — Digital

Digital (119, 245, 120)-net over F₃, using

t-expansion [i] based on digital (113, 245, 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]

(245−126, 245, 810)-Net in Base 3 — Upper bound on s

There is no (119, 245, 811)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 814 474206 335893 688853 103589 215930 410942 809646 209982 540516 629344 423654 256511 441779 245494 917457 408110 370773 530416 928563 > 3²⁴⁵ [i]