MinT - Best Known (139−105, 139, s)-Nets in Base 9

Best Known (139−105, 139, s)-Nets in Base 9

(139−105, 139, 81)-Net over F₉ — Constructive and digital

Digital (34, 139, 81)-net over F₉, using

t-expansion [i] based on digital (32, 139, 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]

(139−105, 139, 128)-Net over F₉ — Digital

Digital (34, 139, 128)-net over F₉, using

t-expansion [i] based on digital (33, 139, 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]

(139−105, 139, 829)-Net in Base 9 — Upper bound on s

There is no (34, 139, 830)-net in base 9, because

1 times m-reduction [i] would yield (34, 138, 830)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 491411 953164 397518 193674 310018 345137 252981 828151 688483 910700 773628 451984 329618 442797 082608 740076 608164 793818 678844 513116 320242 565185 > 9¹³⁸ [i]