MinT - Best Known (139−69, 139, s)-Nets in Base 9

Best Known (139−69, 139, s)-Nets in Base 9

(139−69, 139, 165)-Net over F₉ — Constructive and digital

Digital (70, 139, 165)-net over F₉, using

t-expansion [i] based on digital (64, 139, 165)-net over F₉, using
- net from sequence [i] based on digital (64, 164)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 64 and N(F) ≥ 165, using
    - T₄ from the second tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(139−69, 139, 267)-Net over F₉ — Digital

Digital (70, 139, 267)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(139−69, 139, 12611)-Net in Base 9 — Upper bound on s

There is no (70, 139, 12612)-net in base 9, because

1 times m-reduction [i] would yield (70, 138, 12612)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 485078 318906 138090 195471 228614 558621 102747 491723 109986 429165 964150 418901 337550 175861 513275 561947 583957 882893 011973 389554 138472 069825 > 9¹³⁸ [i]