MinT - Best Known (168−99, 168, s)-Nets in Base 8

Best Known (168−99, 168, s)-Nets in Base 8

(168−99, 168, 98)-Net over F₈ — Constructive and digital

Digital (69, 168, 98)-net over F₈, using

t-expansion [i] based on digital (37, 168, 98)-net over F₈, using
- net from sequence [i] based on digital (37, 97)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 37 and N(F) ≥ 98, using
    - F₅ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]

(168−99, 168, 144)-Net over F₈ — Digital

Digital (69, 168, 144)-net over F₈, using

t-expansion [i] based on digital (45, 168, 144)-net over F₈, using
- net from sequence [i] based on digital (45, 143)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 45 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(168−99, 168, 156)-Net in Base 8

(69, 168, 156)-net in base 8, using

base change [i] based on digital (27, 126, 156)-net over F₁₆, using
- net from sequence [i] based on digital (27, 155)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 27 and N(F) ≥ 156, using
    - a curve from the manYPoints database [i]

(168−99, 168, 3235)-Net in Base 8 — Upper bound on s

There is no (69, 168, 3236)-net in base 8, because

1 times m-reduction [i] would yield (69, 167, 3236)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6 583044 763791 118387 387686 296425 521423 280766 750011 189437 373916 788256 545748 333412 915471 524329 158653 711082 817384 521171 573058 396074 664519 242893 351111 446344 > 8¹⁶⁷ [i]