MinT - Best Known (49, 49+75, s)-Nets in Base 9

Best Known (49, 49+75, s)-Nets in Base 9

(49, 49+75, 81)-Net over F₉ — Constructive and digital

Digital (49, 124, 81)-net over F₉, using

t-expansion [i] based on digital (32, 124, 81)-net over F₉, using
- net from sequence [i] based on digital (32, 80)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
    - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(49, 49+75, 82)-Net in Base 9 — Constructive

(49, 124, 82)-net in base 9, using

2 times m-reduction [i] based on (49, 126, 82)-net in base 9, using
- base change [i] based on digital (7, 84, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]

(49, 49+75, 168)-Net over F₉ — Digital

Digital (49, 124, 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]

(49, 49+75, 2700)-Net in Base 9 — Upper bound on s

There is no (49, 124, 2701)-net in base 9, because

1 times m-reduction [i] would yield (49, 123, 2701)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 2375 296931 575688 100513 558867 988075 857764 933535 552171 069731 532150 379419 789691 552420 969889 799566 056814 632106 322713 407113 > 9¹²³ [i]