MinT - Best Known (12, 31, s)-Nets in Base 3

Best Known (12, 31, s)-Nets in Base 3

(12, 31, 20)-Net over F₃ — Constructive and digital

Digital (12, 31, 20)-net over F₃, using

t-expansion [i] based on digital (11, 31, 20)-net over F₃, using
- net from sequence [i] based on digital (11, 19)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 9, N(F)Â =Â 19, and 1 place with degree 3 [i] based on function field F/F₃ with g(F) = 9 and N(F) ≥ 19, using an explicitly constructive algebraic function field [i]

(12, 31, 22)-Net over F₃ — Digital

Digital (12, 31, 22)-net over F₃, using

net from sequence [i] based on digital (12, 21)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 12 and N(F) ≥ 22, using
  - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(12, 31, 72)-Net in Base 3 — Upper bound on s

There is no (12, 31, 73)-net in base 3, because

1 times m-reduction [i] would yield (12, 30, 73)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 215 182443 828979 > 3³⁰ [i]