MinT - Best Known (167−119, 167, s)-Nets in Base 8

Best Known (167−119, 167, s)-Nets in Base 8

(167−119, 167, 98)-Net over F₈ — Constructive and digital

Digital (48, 167, 98)-net over F₈, using

t-expansion [i] based on digital (37, 167, 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]

(167−119, 167, 144)-Net over F₈ — Digital

Digital (48, 167, 144)-net over F₈, using

t-expansion [i] based on digital (45, 167, 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]

(167−119, 167, 1095)-Net in Base 8 — Upper bound on s

There is no (48, 167, 1096)-net in base 8, because

1 times m-reduction [i] would yield (48, 166, 1096)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 831121 817675 586403 431123 446919 773038 379060 207748 559680 801320 119080 433528 382106 852640 566114 758281 155527 442781 302405 315758 864713 164355 153984 030698 115592 > 8¹⁶⁶ [i]