MinT - Best Known (104−61, 104, s)-Nets in Base 9

Best Known (104−61, 104, s)-Nets in Base 9

(104−61, 104, 81)-Net over F₉ — Constructive and digital

Digital (43, 104, 81)-net over F₉, using

t-expansion [i] based on digital (32, 104, 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]

(104−61, 104, 84)-Net in Base 9 — Constructive

(43, 104, 84)-net in base 9, using

1 times m-reduction [i] based on (43, 105, 84)-net in base 9, using
- base change [i] based on digital (8, 70, 84)-net over F₂₇, using
  - net from sequence [i] based on digital (8, 83)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 8 and N(F) ≥ 84, using
      - a function field by SÃ©mirat [i]

(104−61, 104, 147)-Net over F₉ — Digital

Digital (43, 104, 147)-net over F₉, using

net from sequence [i] based on digital (43, 146)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 43 and N(F) ≥ 147, using
  - a curve from the manYPoints database [i]

(104−61, 104, 2825)-Net in Base 9 — Upper bound on s

There is no (43, 104, 2826)-net in base 9, because

1 times m-reduction [i] would yield (43, 103, 2826)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 194 050197 548993 731830 219911 409413 127778 169198 716674 569838 692656 526802 071705 688057 414596 097032 808097 > 9¹⁰³ [i]