MinT - Best Known (110−63, 110, s)-Nets in Base 9

Best Known (110−63, 110, s)-Nets in Base 9

(110−63, 110, 92)-Net over F₉ — Constructive and digital

Digital (47, 110, 92)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (3, 34, 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 (13, 76, 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]

(110−63, 110, 94)-Net in Base 9 — Constructive

(47, 110, 94)-net in base 9, using

1 times m-reduction [i] based on (47, 111, 94)-net in base 9, using
- base change [i] based on digital (10, 74, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]

(110−63, 110, 162)-Net over F₉ — Digital

Digital (47, 110, 162)-net over F₉, using

t-expansion [i] based on digital (46, 110, 162)-net over F₉, using
- net from sequence [i] based on digital (46, 161)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 46 and N(F) ≥ 162, using
    - a curve from the manYPoints database [i]

(110−63, 110, 3498)-Net in Base 9 — Upper bound on s

There is no (47, 110, 3499)-net in base 9, because

1 times m-reduction [i] would yield (47, 109, 3499)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 103 596702 192120 192199 901916 471670 596443 908790 429978 064710 328549 225796 464536 335047 991392 446147 555786 102313 > 9¹⁰⁹ [i]