MinT - Best Known (100−59, 100, s)-Nets in Base 9

Best Known (100−59, 100, s)-Nets in Base 9

(100−59, 100, 81)-Net over F₉ — Constructive and digital

Digital (41, 100, 81)-net over F₉, using

t-expansion [i] based on digital (32, 100, 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]

(100−59, 100, 82)-Net in Base 9 — Constructive

(41, 100, 82)-net in base 9, using

2 times m-reduction [i] based on (41, 102, 82)-net in base 9, using
- base change [i] based on digital (7, 68, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]

(100−59, 100, 140)-Net over F₉ — Digital

Digital (41, 100, 140)-net over F₉, using

t-expansion [i] based on digital (39, 100, 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]

(100−59, 100, 2622)-Net in Base 9 — Upper bound on s

There is no (41, 100, 2623)-net in base 9, because

1 times m-reduction [i] would yield (41, 99, 2623)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 29675 639283 862151 902162 720310 283835 102736 770437 969335 170192 905883 714379 609109 247965 812330 567961 > 9⁹⁹ [i]