MinT - Best Known (59−31, 59, s)-Nets in Base 3

Best Known (59−31, 59, s)-Nets in Base 3

(59−31, 59, 37)-Net over F₃ — Constructive and digital

Digital (28, 59, 37)-net over F₃, using

t-expansion [i] based on digital (27, 59, 37)-net over F₃, using
- net from sequence [i] based on digital (27, 36)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 26, N(F)Â =Â 36, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using an explicitly constructive algebraic function field [i]

(59−31, 59, 39)-Net over F₃ — Digital

Digital (28, 59, 39)-net over F₃, using

t-expansion [i] based on digital (27, 59, 39)-net over F₃, using
- net from sequence [i] based on digital (27, 38)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 27 and N(F) ≥ 39, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(59−31, 59, 210)-Net in Base 3 — Upper bound on s

There is no (28, 59, 211)-net in base 3, because

1 times m-reduction [i] would yield (28, 58, 211)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 4843 997019 642964 593730 821171 > 3⁵⁸ [i]