MinT - Best Known (231−111, 231, s)-Nets in Base 3

Best Known (231−111, 231, s)-Nets in Base 3

(231−111, 231, 78)-Net over F₃ — Constructive and digital

Digital (120, 231, 78)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (26, 81, 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, 150, 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]

(231−111, 231, 127)-Net over F₃ — Digital

Digital (120, 231, 127)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(231−111, 231, 1001)-Net in Base 3 — Upper bound on s

There is no (120, 231, 1002)-net in base 3, because

1 times m-reduction [i] would yield (120, 230, 1002)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 54 951396 927656 485204 216490 471129 965349 170900 507074 848524 068871 006888 496013 770477 287409 571604 689482 651912 414697 > 3²³⁰ [i]