MinT - Best Known (193−89, 193, s)-Nets in Base 3

Best Known (193−89, 193, s)-Nets in Base 3

(193−89, 193, 76)-Net over F₃ — Constructive and digital

Digital (104, 193, 76)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 59, 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 (45, 134, 48)-net over F₃, using
  - net from sequence [i] based on digital (45, 47)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
      - an explicitly constructive algebraic function field [i]

(193−89, 193, 122)-Net over F₃ — Digital

Digital (104, 193, 122)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(193−89, 193, 999)-Net in Base 3 — Upper bound on s

There is no (104, 193, 1000)-net in base 3, because

1 times m-reduction [i] would yield (104, 192, 1000)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 41 766614 030551 677121 078131 373929 372657 626501 185964 220324 039247 832323 500923 447288 490820 220481 > 3¹⁹² [i]