MinT - Best Known (38, 38+43, s)-Nets in Base 9

Best Known (38, 38+43, s)-Nets in Base 9

(38, 38+43, 94)-Net over F₉ — Constructive and digital

Digital (38, 81, 94)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (4, 25, 30)-net over F₉, using
  - net from sequence [i] based on digital (4, 29)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 4 and N(F) ≥ 30, using
      - a function field by SÃ©mirat [i]
2. digital (13, 56, 64)-net over F₉, using
  - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(38, 38+43, 96)-Net in Base 9 — Constructive

(38, 81, 96)-net in base 9, using

base change [i] based on digital (11, 54, 96)-net over F₂₇, using
- net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
    - an explicitly constructive algebraic function field [i]

(38, 38+43, 130)-Net over F₉ — Digital

Digital (38, 81, 130)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(38, 38+43, 4671)-Net in Base 9 — Upper bound on s

There is no (38, 81, 4672)-net in base 9, because

1 times m-reduction [i] would yield (38, 80, 4672)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 21912 776579 559579 712937 034920 276858 824677 122409 431814 285872 498608 372311 067137 > 9⁸⁰ [i]