MinT - Best Known (166−79, 166, s)-Nets in Base 3

Best Known (166−79, 166, s)-Nets in Base 3

(166−79, 166, 69)-Net over F₃ — Constructive and digital

Digital (87, 166, 69)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (21, 60, 32)-net over F₃, using
  - net from sequence [i] based on digital (21, 31)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 21 and N(F) ≥ 32, using
      - an explicitly constructive algebraic function field [i]
2. digital (27, 106, 37)-net over F₃, using
  - net from sequence [i] based on digital (27, 36)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 26, N(F)Â =Â 36, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using an explicitly constructive algebraic function field [i]

(166−79, 166, 98)-Net over F₃ — Digital

Digital (87, 166, 98)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(166−79, 166, 765)-Net in Base 3 — Upper bound on s

There is no (87, 166, 766)-net in base 3, because

1 times m-reduction [i] would yield (87, 165, 766)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5 374076 247002 004577 429802 212642 726960 359998 579969 433475 973960 398078 151126 441721 > 3¹⁶⁵ [i]