MinT - Best Known (116−67, 116, s)-Nets in Base 9

Best Known (116−67, 116, s)-Nets in Base 9

(116−67, 116, 92)-Net over F₉ — Constructive and digital

Digital (49, 116, 92)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (3, 36, 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, 80, 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]

(116−67, 116, 94)-Net in Base 9 — Constructive

(49, 116, 94)-net in base 9, using

1 times m-reduction [i] based on (49, 117, 94)-net in base 9, using
- base change [i] based on digital (10, 78, 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]

(116−67, 116, 168)-Net over F₉ — Digital

Digital (49, 116, 168)-net over F₉, using

net from sequence [i] based on digital (49, 167)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 49 and N(F) ≥ 168, using
  - Drinfeld modules of rank 1 [i]

(116−67, 116, 3460)-Net in Base 9 — Upper bound on s

There is no (49, 116, 3461)-net in base 9, because

1 times m-reduction [i] would yield (49, 115, 3461)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 54 940192 872314 897123 592684 309736 744467 994893 216604 589665 095550 891845 825780 318112 436301 631737 498970 158829 636905 > 9¹¹⁵ [i]