MinT - Best Known (126−103, 126, s)-Nets in Base 9

Best Known (126−103, 126, s)-Nets in Base 9

(126−103, 126, 78)-Net over F₉ — Constructive and digital

Digital (23, 126, 78)-net over F₉, using

t-expansion [i] based on digital (22, 126, 78)-net over F₉, using
- net from sequence [i] based on digital (22, 77)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
    - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(126−103, 126, 92)-Net over F₉ — Digital

Digital (23, 126, 92)-net over F₉, using

net from sequence [i] based on digital (23, 91)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 23 and N(F) ≥ 92, using
  - Drinfeld modules of rank 1 [i]

(126−103, 126, 510)-Net in Base 9 — Upper bound on s

There is no (23, 126, 511)-net in base 9, because

1 times m-reduction [i] would yield (23, 125, 511)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 197253 125224 736032 015359 769008 200691 654797 466050 644854 792322 361899 806040 552379 900055 330865 817027 655817 554505 067906 625129 > 9¹²⁵ [i]