MinT - Best Known (219−99, 219, s)-Nets in Base 3

Best Known (219−99, 219, s)-Nets in Base 3

(219−99, 219, 84)-Net over F₃ — Constructive and digital

Digital (120, 219, 84)-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 (45, 144, 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]

(219−99, 219, 144)-Net over F₃ — Digital

Digital (120, 219, 144)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−99, 219, 1219)-Net in Base 3 — Upper bound on s

There is no (120, 219, 1220)-net in base 3, because

1 times m-reduction [i] would yield (120, 218, 1220)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 103 484992 449200 977221 960901 704044 126822 063366 149601 126755 001624 781876 804032 685538 842999 899595 529980 521097 > 3²¹⁸ [i]