MinT - Best Known (172−95, 172, s)-Nets in Base 3

Best Known (172−95, 172, s)-Nets in Base 3

(172−95, 172, 56)-Net over F₃ — Constructive and digital

Digital (77, 172, 56)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 62, 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 (15, 110, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(172−95, 172, 84)-Net over F₃ — Digital

Digital (77, 172, 84)-net over F₃, using

t-expansion [i] based on digital (71, 172, 84)-net over F₃, using
- net from sequence [i] based on digital (71, 83)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 71 and N(F) ≥ 84, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(172−95, 172, 455)-Net in Base 3 — Upper bound on s

There is no (77, 172, 456)-net in base 3, because

1 times m-reduction [i] would yield (77, 171, 456)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 4162 993844 304016 763709 248257 220760 578473 352868 297628 784146 893724 304317 415466 853217 > 3¹⁷¹ [i]