MinT - Best Known (49, 49+73, s)-Nets in Base 9

Best Known (49, 49+73, s)-Nets in Base 9

(49, 49+73, 81)-Net over F₉ — Constructive and digital

Digital (49, 122, 81)-net over F₉, using

t-expansion [i] based on digital (32, 122, 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+73, 84)-Net in Base 9 — Constructive

(49, 122, 84)-net in base 9, using

1 times m-reduction [i] based on (49, 123, 84)-net in base 9, using
- base change [i] based on digital (8, 82, 84)-net over F₂₇, using
  - net from sequence [i] based on digital (8, 83)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 8 and N(F) ≥ 84, using
      - a function field by SÃ©mirat [i]

(49, 49+73, 168)-Net over F₉ — Digital

Digital (49, 122, 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+73, 2855)-Net in Base 9 — Upper bound on s

There is no (49, 122, 2856)-net in base 9, because

1 times m-reduction [i] would yield (49, 121, 2856)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 29 356517 885985 194868 623741 732977 676364 330244 135140 747041 701170 266349 541884 748385 948068 843836 336505 503678 972789 795585 > 9¹²¹ [i]