MinT - Best Known (113−71, 113, s)-Nets in Base 9

Best Known (113−71, 113, s)-Nets in Base 9

(113−71, 113, 81)-Net over F₉ — Constructive and digital

Digital (42, 113, 81)-net over F₉, using

t-expansion [i] based on digital (32, 113, 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]

(113−71, 113, 140)-Net over F₉ — Digital

Digital (42, 113, 140)-net over F₉, using

t-expansion [i] based on digital (39, 113, 140)-net over F₉, using
- net from sequence [i] based on digital (39, 139)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 39 and N(F) ≥ 140, using
    - a curve from the manYPoints database [i]

(113−71, 113, 1945)-Net in Base 9 — Upper bound on s

There is no (42, 113, 1946)-net in base 9, because

1 times m-reduction [i] would yield (42, 112, 1946)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 75760 086162 366303 025292 638822 723201 527081 309685 159544 268318 050536 144571 409494 667916 247756 773862 946701 568945 > 9¹¹² [i]