MinT - Best Known (63, 63+85, s)-Nets in Base 9

Best Known (63, 63+85, s)-Nets in Base 9

(63, 63+85, 104)-Net over F₉ — Constructive and digital

Digital (63, 148, 104)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (8, 50, 40)-net over F₉, using
  - net from sequence [i] based on digital (8, 39)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 8 and N(F) ≥ 40, using
      - a function field by SÃ©mirat [i]
2. digital (13, 98, 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]

(63, 63+85, 192)-Net over F₉ — Digital

Digital (63, 148, 192)-net over F₉, using

t-expansion [i] based on digital (61, 148, 192)-net over F₉, using
- net from sequence [i] based on digital (61, 191)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 61 and N(F) ≥ 192, using
    - Drinfeld modules of rank 1 [i]

(63, 63+85, 4488)-Net in Base 9 — Upper bound on s

There is no (63, 148, 4489)-net in base 9, because

1 times m-reduction [i] would yield (63, 147, 4489)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 189 035437 473963 573702 019553 615320 604484 528435 239860 787420 223088 064714 840671 862342 735415 926734 089218 797585 467537 525450 252621 336336 859676 604945 > 9¹⁴⁷ [i]