MinT - Best Known (68, 68+81, s)-Nets in Base 3

Best Known (68, 68+81, s)-Nets in Base 3

(68, 68+81, 52)-Net over F₃ — Constructive and digital

Digital (68, 149, 52)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (13, 53, 24)-net over F₃, using
  - net from sequence [i] based on digital (13, 23)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 13 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [i]
2. digital (15, 96, 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]

(68, 68+81, 72)-Net over F₃ — Digital

Digital (68, 149, 72)-net over F₃, using

t-expansion [i] based on digital (67, 149, 72)-net over F₃, using
- net from sequence [i] based on digital (67, 71)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 67 and N(F) ≥ 72, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(68, 68+81, 421)-Net in Base 3 — Upper bound on s

There is no (68, 149, 422)-net in base 3, because

1 times m-reduction [i] would yield (68, 148, 422)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 44205 377691 534751 194613 082013 129262 930022 484097 633008 423447 474660 491025 > 3¹⁴⁸ [i]