MinT - Best Known (137−107, 137, s)-Nets in Base 9

Best Known (137−107, 137, s)-Nets in Base 9

(137−107, 137, 78)-Net over F₉ — Constructive and digital

Digital (30, 137, 78)-net over F₉, using

t-expansion [i] based on digital (22, 137, 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]

(137−107, 137, 110)-Net over F₉ — Digital

Digital (30, 137, 110)-net over F₉, using

t-expansion [i] based on digital (26, 137, 110)-net over F₉, using
- net from sequence [i] based on digital (26, 109)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 26 and N(F) ≥ 110, using
    - a curve from the manYPoints database [i]

(137−107, 137, 691)-Net in Base 9 — Upper bound on s

There is no (30, 137, 692)-net in base 9, because

1 times m-reduction [i] would yield (30, 136, 692)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 6389 327200 705803 010828 604664 979379 320666 554123 304241 313986 895435 229886 036027 754108 557912 136957 844793 567983 646620 902226 228007 959329 > 9¹³⁶ [i]