MinT - Best Known (112−69, 112, s)-Nets in Base 9

Best Known (112−69, 112, s)-Nets in Base 9

(112−69, 112, 81)-Net over F₉ — Constructive and digital

Digital (43, 112, 81)-net over F₉, using

t-expansion [i] based on digital (32, 112, 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]

(112−69, 112, 147)-Net over F₉ — Digital

Digital (43, 112, 147)-net over F₉, using

net from sequence [i] based on digital (43, 146)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 43 and N(F) ≥ 147, using
  - a curve from the manYPoints database [i]

(112−69, 112, 2185)-Net in Base 9 — Upper bound on s

There is no (43, 112, 2186)-net in base 9, because

1 times m-reduction [i] would yield (43, 111, 2186)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 8358 297692 918540 881653 004506 351579 781029 785324 229950 086169 884866 705976 078710 600905 187434 052656 919128 900705 > 9¹¹¹ [i]