MinT - Best Known (65−33, 65, s)-Nets in Base 3

Best Known (65−33, 65, s)-Nets in Base 3

(65−33, 65, 38)-Net over F₃ — Constructive and digital

Digital (32, 65, 38)-net over F₃, using

net from sequence [i] based on digital (32, 37)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 32 and N(F) ≥ 38, using
  - an explicitly constructive algebraic function field [i]

(65−33, 65, 42)-Net over F₃ — Digital

Digital (32, 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]

(65−33, 65, 260)-Net in Base 3 — Upper bound on s

There is no (32, 65, 261)-net in base 3, because

1 times m-reduction [i] would yield (32, 64, 261)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3 581647 086406 004612 137635 554753 > 3⁶⁴ [i]