MinT - Best Known (131−83, 131, s)-Nets in Base 9

Best Known (131−83, 131, s)-Nets in Base 9

(131−83, 131, 81)-Net over F₉ — Constructive and digital

Digital (48, 131, 81)-net over F₉, using

t-expansion [i] based on digital (32, 131, 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]

(131−83, 131, 163)-Net over F₉ — Digital

Digital (48, 131, 163)-net over F₉, using

net from sequence [i] based on digital (48, 162)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 48 and N(F) ≥ 163, using
  - Drinfeld modules of rank 1 [i]

(131−83, 131, 2115)-Net in Base 9 — Upper bound on s

There is no (48, 131, 2116)-net in base 9, because

1 times m-reduction [i] would yield (48, 130, 2116)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 11425 284372 842755 570816 339670 354130 340453 502284 658313 440073 261403 508538 585848 908898 258298 006273 531694 586410 300919 192196 675873 > 9¹³⁰ [i]