MinT - Best Known (160−75, 160, s)-Nets in Base 3

Best Known (160−75, 160, s)-Nets in Base 3

(160−75, 160, 69)-Net over F₃ — Constructive and digital

Digital (85, 160, 69)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (21, 58, 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, 102, 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]

(160−75, 160, 99)-Net over F₃ — Digital

Digital (85, 160, 99)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(160−75, 160, 786)-Net in Base 3 — Upper bound on s

There is no (85, 160, 787)-net in base 3, because

1 times m-reduction [i] would yield (85, 159, 787)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7345 920438 647378 153906 237988 835117 298329 864933 157522 521200 457297 116576 630495 > 3¹⁵⁹ [i]