MinT - Best Known (137−71, 137, s)-Nets in Base 9

Best Known (137−71, 137, s)-Nets in Base 9

(137−71, 137, 165)-Net over F₉ — Constructive and digital

Digital (66, 137, 165)-net over F₉, using

t-expansion [i] based on digital (64, 137, 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]

(137−71, 137, 220)-Net over F₉ — Digital

Digital (66, 137, 220)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(137−71, 137, 8851)-Net in Base 9 — Upper bound on s

There is no (66, 137, 8852)-net in base 9, because

1 times m-reduction [i] would yield (66, 136, 8852)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 5984 118340 220529 305078 866017 284272 254338 777730 713346 012186 693683 222378 144487 615366 516027 885501 962133 722150 458926 957924 143617 167201 > 9¹³⁶ [i]