MinT - Best Known (149−45, 149, s)-Nets in Base 9

Best Known (149−45, 149, s)-Nets in Base 9

(149−45, 149, 772)-Net over F₉ — Constructive and digital

Digital (104, 149, 772)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (5, 27, 32)-net over F₉, using
  - net from sequence [i] based on digital (5, 31)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 5 and N(F) ≥ 32, using
      - a function field by GarcÃa and Quoos [i]
2. digital (77, 122, 740)-net over F₉, using
  - trace code for nets [i] based on digital (16, 61, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(149−45, 149, 3697)-Net over F₉ — Digital

Digital (104, 149, 3697)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(149−45, 149, 2973108)-Net in Base 9 — Upper bound on s

There is no (104, 149, 2973109)-net in base 9, because

1 times m-reduction [i] would yield (104, 148, 2973109)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 1690 028376 414746 995284 858700 529283 689962 075120 779766 653457 313385 335503 891649 678164 017602 891835 045238 004708 411893 498973 125902 838397 244745 049265 > 9¹⁴⁸ [i]