MinT - Best Known (137−69, 137, s)-Nets in Base 9

Best Known (137−69, 137, s)-Nets in Base 9

(137−69, 137, 165)-Net over F₉ — Constructive and digital

Digital (68, 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]

(137−69, 137, 248)-Net over F₉ — Digital

Digital (68, 137, 248)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(137−69, 137, 11080)-Net in Base 9 — Upper bound on s

There is no (68, 137, 11081)-net in base 9, because

1 times m-reduction [i] would yield (68, 136, 11081)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 6000 519037 129137 607516 187602 588761 760042 349853 297340 894828 104116 821097 344685 045128 005599 244072 253124 673344 254225 738412 006407 478481 > 9¹³⁶ [i]