MinT - Best Known (159−107, 159, s)-Nets in Base 8

Best Known (159−107, 159, s)-Nets in Base 8

(159−107, 159, 98)-Net over F₈ — Constructive and digital

Digital (52, 159, 98)-net over F₈, using

t-expansion [i] based on digital (37, 159, 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]

(159−107, 159, 144)-Net over F₈ — Digital

Digital (52, 159, 144)-net over F₈, using

t-expansion [i] based on digital (45, 159, 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]

(159−107, 159, 1415)-Net in Base 8 — Upper bound on s

There is no (52, 159, 1416)-net in base 8, because

1 times m-reduction [i] would yield (52, 158, 1416)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 50264 473218 206385 053974 730172 931219 186408 145057 464382 627320 632653 097049 481472 744117 126849 771254 486074 614175 887064 276346 866874 990597 718849 461352 > 8¹⁵⁸ [i]