MinT - Best Known (151−69, 151, s)-Nets in Base 3

Best Known (151−69, 151, s)-Nets in Base 3

(151−69, 151, 69)-Net over F₃ — Constructive and digital

Digital (82, 151, 69)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (21, 55, 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, 96, 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]

(151−69, 151, 101)-Net over F₃ — Digital

Digital (82, 151, 101)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(151−69, 151, 828)-Net in Base 3 — Upper bound on s

There is no (82, 151, 829)-net in base 3, because

1 times m-reduction [i] would yield (82, 150, 829)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 372862 484355 729865 540770 787883 299716 846765 990315 187983 577396 831774 841601 > 3¹⁵⁰ [i]