MinT - Best Known (117−93, 117, s)-Nets in Base 9

Best Known (117−93, 117, s)-Nets in Base 9

(117−93, 117, 78)-Net over F₉ — Constructive and digital

Digital (24, 117, 78)-net over F₉, using

t-expansion [i] based on digital (22, 117, 78)-net over F₉, using
- net from sequence [i] based on digital (22, 77)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
    - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(117−93, 117, 92)-Net over F₉ — Digital

Digital (24, 117, 92)-net over F₉, using

t-expansion [i] based on digital (23, 117, 92)-net over F₉, using
- net from sequence [i] based on digital (23, 91)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 23 and N(F) ≥ 92, using
    - Drinfeld modules of rank 1 [i]

(117−93, 117, 545)-Net in Base 9 — Upper bound on s

There is no (24, 117, 546)-net in base 9, because

1 times m-reduction [i] would yield (24, 116, 546)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 508 528281 811368 131805 171355 676865 707350 887042 497502 401225 190331 613087 551138 949617 885271 716846 527238 075308 729633 > 9¹¹⁶ [i]