MinT - Best Known (56, 56+61, s)-Nets in Base 9

Best Known (56, 56+61, s)-Nets in Base 9

(56, 56+61, 128)-Net over F₉ — Constructive and digital

Digital (56, 117, 128)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (13, 43, 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]
2. digital (13, 74, 64)-net over F₉, using
  - net from sequence [i] based on digital (13, 63)-sequence over F₉ (see above)

(56, 56+61, 188)-Net over F₉ — Digital

Digital (56, 117, 188)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(56, 56+61, 7351)-Net in Base 9 — Upper bound on s

There is no (56, 117, 7352)-net in base 9, because

1 times m-reduction [i] would yield (56, 116, 7352)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 493 722005 834367 774362 959420 575601 776941 950742 880793 738658 394175 766224 645152 547938 853187 881527 390292 312024 234369 > 9¹¹⁶ [i]