MinT - Best Known (158−103, 158, s)-Nets in Base 8

Best Known (158−103, 158, s)-Nets in Base 8

(158−103, 158, 98)-Net over F₈ — Constructive and digital

Digital (55, 158, 98)-net over F₈, using

t-expansion [i] based on digital (37, 158, 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]

(158−103, 158, 144)-Net over F₈ — Digital

Digital (55, 158, 144)-net over F₈, using

t-expansion [i] based on digital (45, 158, 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]

(158−103, 158, 1677)-Net in Base 8 — Upper bound on s

There is no (55, 158, 1678)-net in base 8, because

1 times m-reduction [i] would yield (55, 157, 1678)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6215 544366 013953 942468 153777 686353 303430 264835 313838 263843 495127 795790 064203 673720 343608 058700 918201 783892 494474 748811 674607 815641 464270 510384 > 8¹⁵⁷ [i]