MinT - Best Known (107−63, 107, s)-Nets in Base 9

Best Known (107−63, 107, s)-Nets in Base 9

(107−63, 107, 81)-Net over F₉ — Constructive and digital

Digital (44, 107, 81)-net over F₉, using

t-expansion [i] based on digital (32, 107, 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]

(107−63, 107, 84)-Net in Base 9 — Constructive

(44, 107, 84)-net in base 9, using

1 times m-reduction [i] based on (44, 108, 84)-net in base 9, using
- base change [i] based on digital (8, 72, 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]

(107−63, 107, 147)-Net over F₉ — Digital

Digital (44, 107, 147)-net over F₉, using

t-expansion [i] based on digital (43, 107, 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]

(107−63, 107, 2824)-Net in Base 9 — Upper bound on s

There is no (44, 107, 2825)-net in base 9, because

1 times m-reduction [i] would yield (44, 106, 2825)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 141846 443743 616911 264045 891393 788335 613346 863129 537889 898165 576325 625738 902727 722198 066881 472834 012665 > 9¹⁰⁶ [i]