MinT - Best Known (96, 96+41, s)-Nets in Base 9

Best Known (96, 96+41, s)-Nets in Base 9

(96, 96+41, 768)-Net over F₉ — Constructive and digital

Digital (96, 137, 768)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (3, 23, 28)-net over F₉, using
  - net from sequence [i] based on digital (3, 27)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 3 and N(F) ≥ 28, using
      - the Hermitian function field over F₉ [i]
2. digital (73, 114, 740)-net over F₉, using
  - trace code for nets [i] based on digital (16, 57, 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]

(96, 96+41, 3676)-Net over F₉ — Digital

Digital (96, 137, 3676)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(96, 96+41, 3199364)-Net in Base 9 — Upper bound on s

There is no (96, 137, 3199365)-net in base 9, because

1 times m-reduction [i] would yield (96, 136, 3199365)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 5983 888472 277441 769204 597853 564845 306094 016878 965690 185421 596154 183308 571530 163312 741674 260169 561110 431386 828923 489433 033683 383201 > 9¹³⁶ [i]