MinT - Best Known (110−57, 110, s)-Nets in Base 9

Best Known (110−57, 110, s)-Nets in Base 9

(110−57, 110, 114)-Net over F₉ — Constructive and digital

Digital (53, 110, 114)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (8, 36, 40)-net over F₉, using
  - net from sequence [i] based on digital (8, 39)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 8 and N(F) ≥ 40, using
      - a function field by SÃ©mirat [i]
2. digital (17, 74, 74)-net over F₉, using
  - net from sequence [i] based on digital (17, 73)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
      - a function field by GarcÃa and Quoos [i]

(110−57, 110, 183)-Net over F₉ — Digital

Digital (53, 110, 183)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(110−57, 110, 7305)-Net in Base 9 — Upper bound on s

There is no (53, 110, 7306)-net in base 9, because

1 times m-reduction [i] would yield (53, 109, 7306)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 103 193498 284139 950520 699400 716908 204878 203980 555888 342761 538070 674088 888190 921822 835208 471498 710672 053825 > 9¹⁰⁹ [i]