MinT - Best Known (145−69, 145, s)-Nets in Base 3

Best Known (145−69, 145, s)-Nets in Base 3

(145−69, 145, 65)-Net over F₃ — Constructive and digital

Digital (76, 145, 65)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 49, 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]
2. digital (27, 96, 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]

(145−69, 145, 87)-Net over F₃ — Digital

Digital (76, 145, 87)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(145−69, 145, 677)-Net in Base 3 — Upper bound on s

There is no (76, 145, 678)-net in base 3, because

1 times m-reduction [i] would yield (76, 144, 678)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 531 446806 599402 176624 825191 557060 153763 275750 411344 020266 922029 505389 > 3¹⁴⁴ [i]