MinT - Best Known (116−68, 116, s)-Nets in Base 9

Best Known (116−68, 116, s)-Nets in Base 9

(116−68, 116, 81)-Net over F₉ — Constructive and digital

Digital (48, 116, 81)-net over F₉, using

t-expansion [i] based on digital (32, 116, 81)-net over F₉, using
- net from sequence [i] based on digital (32, 80)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
    - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(116−68, 116, 88)-Net in Base 9 — Constructive

(48, 116, 88)-net in base 9, using

1 times m-reduction [i] based on (48, 117, 88)-net in base 9, using
- base change [i] based on digital (9, 78, 88)-net over F₂₇, using
  - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
      - a function field by SÃ©mirat [i]

(116−68, 116, 163)-Net over F₉ — Digital

Digital (48, 116, 163)-net over F₉, using

net from sequence [i] based on digital (48, 162)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 48 and N(F) ≥ 163, using
  - Drinfeld modules of rank 1 [i]

(116−68, 116, 3027)-Net in Base 9 — Upper bound on s

There is no (48, 116, 3028)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 495 262411 583729 845074 250303 840581 548574 475761 518813 249947 708580 188429 692635 033475 204328 476532 709381 628397 671873 > 9¹¹⁶ [i]