MinT - Best Known (166−95, 166, s)-Nets in Base 8

Best Known (166−95, 166, s)-Nets in Base 8

(166−95, 166, 111)-Net over F₈ — Constructive and digital

Digital (71, 166, 111)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (10, 57, 46)-net over F₈, using
  - net from sequence [i] based on digital (10, 45)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₈ with g(F)Â =Â 9, N(F)Â =Â 45, and 1 place with degree 2 [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using an explicitly constructive algebraic function field [i]
2. digital (14, 109, 65)-net over F₈, using
  - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
      - the Suzuki function field over F₈ [i]

(166−95, 166, 155)-Net over F₈ — Digital

Digital (71, 166, 155)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(166−95, 166, 161)-Net in Base 8

(71, 166, 161)-net in base 8, using

2 times m-reduction [i] based on (71, 168, 161)-net in base 8, using
- base change [i] based on digital (29, 126, 161)-net over F₁₆, using
  - net from sequence [i] based on digital (29, 160)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 29 and N(F) ≥ 161, using
      - a curve from the manYPoints database [i]

(166−95, 166, 3855)-Net in Base 8 — Upper bound on s

There is no (71, 166, 3856)-net in base 8, because

1 times m-reduction [i] would yield (71, 165, 3856)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 102732 802950 504781 672480 820659 817622 123660 617358 727736 497133 795456 733277 927723 201561 960377 938169 361829 260055 496541 788493 091766 075490 642250 365765 440408 > 8¹⁶⁵ [i]