MinT - Best Known (149−89, 149, s)-Nets in Base 9

Best Known (149−89, 149, s)-Nets in Base 9

(149−89, 149, 92)-Net over F₉ — Constructive and digital

Digital (60, 149, 92)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (3, 47, 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, 102, 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]

(149−89, 149, 94)-Net in Base 9 — Constructive

(60, 149, 94)-net in base 9, using

1 times m-reduction [i] based on (60, 150, 94)-net in base 9, using
- base change [i] based on digital (10, 100, 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]

(149−89, 149, 190)-Net over F₉ — Digital

Digital (60, 149, 190)-net over F₉, using

net from sequence [i] based on digital (60, 189)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 60 and N(F) ≥ 190, using
  - Drinfeld modules of rank 1 [i]

(149−89, 149, 3468)-Net in Base 9 — Upper bound on s

There is no (60, 149, 3469)-net in base 9, because

1 times m-reduction [i] would yield (60, 148, 3469)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 1691 101642 539587 398044 603452 775918 295214 415023 579848 406933 828775 081607 396970 232001 058928 994408 009905 941028 494831 147573 779253 371420 173999 233889 > 9¹⁴⁸ [i]