MinT - Best Known (147−87, 147, s)-Nets in Base 9

Best Known (147−87, 147, s)-Nets in Base 9

(147−87, 147, 94)-Net over F₉ — Constructive and digital

Digital (60, 147, 94)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (4, 47, 30)-net over F₉, using
  - net from sequence [i] based on digital (4, 29)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 4 and N(F) ≥ 30, using
      - a function field by SÃ©mirat [i]
2. digital (13, 100, 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]

(147−87, 147, 96)-Net in Base 9 — Constructive

(60, 147, 96)-net in base 9, using

base change [i] based on digital (11, 98, 96)-net over F₂₇, using
- net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
    - an explicitly constructive algebraic function field [i]

(147−87, 147, 190)-Net over F₉ — Digital

Digital (60, 147, 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]

(147−87, 147, 3641)-Net in Base 9 — Upper bound on s

There is no (60, 147, 3642)-net in base 9, because

1 times m-reduction [i] would yield (60, 146, 3642)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 21 077091 512585 666711 252991 856261 903533 945945 699540 757441 536911 277249 501153 633126 293175 484789 139858 507235 254398 624464 639653 592213 594224 937009 > 9¹⁴⁶ [i]