MinT - Best Known (122−35, 122, s)-Nets in Base 9

Best Known (122−35, 122, s)-Nets in Base 9

(122−35, 122, 768)-Net over F₉ — Constructive and digital

Digital (87, 122, 768)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (3, 20, 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 (67, 102, 740)-net over F₉, using
  - trace code for nets [i] based on digital (16, 51, 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]

(122−35, 122, 4509)-Net over F₉ — Digital

Digital (87, 122, 4509)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(122−35, 122, 5556685)-Net in Base 9 — Upper bound on s

There is no (87, 122, 5556686)-net in base 9, because

1 times m-reduction [i] would yield (87, 121, 5556686)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 29 063256 244319 013373 187931 324261 834308 233807 865726 052225 221436 519396 568403 412217 141164 143204 794657 047694 589427 181425 > 9¹²¹ [i]