MinT - Best Known (213−99, 213, s)-Nets in Base 3

Best Known (213−99, 213, s)-Nets in Base 3

(213−99, 213, 78)-Net over F₃ — Constructive and digital

Digital (114, 213, 78)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (26, 75, 36)-net over F₃, using
  - net from sequence [i] based on digital (26, 35)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using
      - an explicitly constructive algebraic function field [i]
2. digital (39, 138, 42)-net over F₃, using
  - net from sequence [i] based on digital (39, 41)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 39 and N(F) ≥ 42, using
      - an explicitly constructive algebraic function field [i]

(213−99, 213, 130)-Net over F₃ — Digital

Digital (114, 213, 130)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(213−99, 213, 1060)-Net in Base 3 — Upper bound on s

There is no (114, 213, 1061)-net in base 3, because

1 times m-reduction [i] would yield (114, 212, 1061)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 144884 062742 408364 342446 609927 929031 123476 003942 240376 920381 160220 150729 885380 297772 626624 970101 683083 > 3²¹² [i]