MinT - Best Known (130−81, 130, s)-Nets in Base 9

Best Known (130−81, 130, s)-Nets in Base 9

(130−81, 130, 81)-Net over F₉ — Constructive and digital

Digital (49, 130, 81)-net over F₉, using

t-expansion [i] based on digital (32, 130, 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]

(130−81, 130, 168)-Net over F₉ — Digital

Digital (49, 130, 168)-net over F₉, using

net from sequence [i] based on digital (49, 167)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 49 and N(F) ≥ 168, using
  - Drinfeld modules of rank 1 [i]

(130−81, 130, 2331)-Net in Base 9 — Upper bound on s

There is no (49, 130, 2332)-net in base 9, because

1 times m-reduction [i] would yield (49, 129, 2332)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 1265 035969 408804 383210 819685 471360 150367 811036 615429 870069 227464 855786 595163 409096 026624 653138 585461 305097 137897 535608 866049 > 9¹²⁹ [i]