MinT - Best Known (154−75, 154, s)-Nets in Base 3

Best Known (154−75, 154, s)-Nets in Base 3

(154−75, 154, 65)-Net over F₃ — Constructive and digital

Digital (79, 154, 65)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 52, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, 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]

(154−75, 154, 86)-Net over F₃ — Digital

Digital (79, 154, 86)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(154−75, 154, 652)-Net in Base 3 — Upper bound on s

There is no (79, 154, 653)-net in base 3, because

1 times m-reduction [i] would yield (79, 153, 653)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 10 158837 034125 169321 846981 496335 403320 226425 869419 597098 645979 311964 692395 > 3¹⁵³ [i]