MinT - Best Known (127−77, 127, s)-Nets in Base 9

Best Known (127−77, 127, s)-Nets in Base 9

(127−77, 127, 81)-Net over F₉ — Constructive and digital

Digital (50, 127, 81)-net over F₉, using

t-expansion [i] based on digital (32, 127, 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]

(127−77, 127, 82)-Net in Base 9 — Constructive

(50, 127, 82)-net in base 9, using

2 times m-reduction [i] based on (50, 129, 82)-net in base 9, using
- base change [i] based on digital (7, 86, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]

(127−77, 127, 182)-Net over F₉ — Digital

Digital (50, 127, 182)-net over F₉, using

net from sequence [i] based on digital (50, 181)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 50 and N(F) ≥ 182, using
  - a curve from the manYPoints database [i]

(127−77, 127, 2716)-Net in Base 9 — Upper bound on s

There is no (50, 127, 2717)-net in base 9, because

1 times m-reduction [i] would yield (50, 126, 2717)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 1 716239 424278 109828 008836 794501 676814 338756 555701 825345 040981 223749 161145 924778 305506 225485 962406 722622 275966 639685 952945 > 9¹²⁶ [i]