MinT - Best Known (51, 51+53, s)-Nets in Base 9

Best Known (51, 51+53, s)-Nets in Base 9

(51, 51+53, 114)-Net over F₉ — Constructive and digital

Digital (51, 104, 114)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (8, 34, 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, 70, 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]

(51, 51+53, 188)-Net over F₉ — Digital

Digital (51, 104, 188)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(51, 51+53, 7936)-Net in Base 9 — Upper bound on s

There is no (51, 104, 7937)-net in base 9, because

1 times m-reduction [i] would yield (51, 103, 7937)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 194 159235 326123 644224 533632 850949 071024 983667 879297 395525 064894 528096 519146 006074 368554 998833 893905 > 9¹⁰³ [i]