MinT - Best Known (141−103, 141, s)-Nets in Base 9

Best Known (141−103, 141, s)-Nets in Base 9

(141−103, 141, 81)-Net over F₉ — Constructive and digital

Digital (38, 141, 81)-net over F₉, using

t-expansion [i] based on digital (32, 141, 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]

(141−103, 141, 128)-Net over F₉ — Digital

Digital (38, 141, 128)-net over F₉, using

t-expansion [i] based on digital (33, 141, 128)-net over F₉, using
- net from sequence [i] based on digital (33, 127)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 33 and N(F) ≥ 128, using
    - a curve from the manYPoints database [i]

(141−103, 141, 1002)-Net in Base 9 — Upper bound on s

There is no (38, 141, 1003)-net in base 9, because

1 times m-reduction [i] would yield (38, 140, 1003)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 40 704062 416648 515568 358991 281569 152615 563935 182472 877838 893775 006553 934131 831192 427134 742816 502667 687910 530391 037290 778078 027409 929225 > 9¹⁴⁰ [i]