MinT - Best Known (51, 51+76, s)-Nets in Base 9

Best Known (51, 51+76, s)-Nets in Base 9

(51, 51+76, 81)-Net over F₉ — Constructive and digital

Digital (51, 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]

(51, 51+76, 84)-Net in Base 9 — Constructive

(51, 127, 84)-net in base 9, using

2 times m-reduction [i] based on (51, 129, 84)-net in base 9, using
- base change [i] based on digital (8, 86, 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]

(51, 51+76, 182)-Net over F₉ — Digital

Digital (51, 127, 182)-net over F₉, using

t-expansion [i] based on 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]

(51, 51+76, 2880)-Net in Base 9 — Upper bound on s

There is no (51, 127, 2881)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 15 627189 443239 743083 488565 132648 490765 948429 612594 378238 317018 182224 578274 204515 800670 568158 834987 970583 000750 359115 869425 > 9¹²⁷ [i]