MinT - Best Known (138−107, 138, s)-Nets in Base 9

Best Known (138−107, 138, s)-Nets in Base 9

(138−107, 138, 78)-Net over F₉ — Constructive and digital

Digital (31, 138, 78)-net over F₉, using

t-expansion [i] based on digital (22, 138, 78)-net over F₉, using
- net from sequence [i] based on digital (22, 77)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
    - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(138−107, 138, 120)-Net over F₉ — Digital

Digital (31, 138, 120)-net over F₉, using

net from sequence [i] based on digital (31, 119)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 31 and N(F) ≥ 120, using
  - a curve from the manYPoints database [i]

(138−107, 138, 721)-Net in Base 9 — Upper bound on s

There is no (31, 138, 722)-net in base 9, because

1 times m-reduction [i] would yield (31, 137, 722)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 54984 311485 383105 944505 372066 313301 395646 094607 262707 482429 124904 869695 318687 966284 134985 574122 378388 658330 172298 672485 705368 967505 > 9¹³⁷ [i]