MinT - Best Known (138−77, 138, s)-Nets in Base 9

Best Known (138−77, 138, s)-Nets in Base 9

(138−77, 138, 108)-Net over F₉ — Constructive and digital

Digital (61, 138, 108)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (6, 44, 34)-net over F₉, using
  - net from sequence [i] based on digital (6, 33)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 6 and N(F) ≥ 34, using
      - a function field by SÃ©mirat [i]
2. digital (17, 94, 74)-net over F₉, using
  - net from sequence [i] based on digital (17, 73)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
      - a function field by GarcÃa and Quoos [i]

(138−77, 138, 192)-Net over F₉ — Digital

Digital (61, 138, 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]

(138−77, 138, 5152)-Net in Base 9 — Upper bound on s

There is no (61, 138, 5153)-net in base 9, because

1 times m-reduction [i] would yield (61, 137, 5153)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 53874 163450 486706 374878 532292 726227 589543 360995 102896 235423 011699 771171 354246 249149 519675 428362 337454 034346 688241 192579 606807 110385 > 9¹³⁷ [i]