MinT - Best Known (65−35, 65, s)-Nets in Base 3

Best Known (65−35, 65, s)-Nets in Base 3

(65−35, 65, 37)-Net over F₃ — Constructive and digital

Digital (30, 65, 37)-net over F₃, using

t-expansion [i] based on digital (27, 65, 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]

(65−35, 65, 42)-Net over F₃ — Digital

Digital (30, 65, 42)-net over F₃, using

t-expansion [i] based on digital (29, 65, 42)-net over F₃, using
- net from sequence [i] based on digital (29, 41)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 29 and N(F) ≥ 42, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(65−35, 65, 208)-Net in Base 3 — Upper bound on s

There is no (30, 65, 209)-net in base 3, because

1 times m-reduction [i] would yield (30, 64, 209)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 580520 218377 866600 450588 736867 > 3⁶⁴ [i]