MinT - Best Known (179−90, 179, s)-Nets in Base 3

Best Known (179−90, 179, s)-Nets in Base 3

(179−90, 179, 65)-Net over F₃ — Constructive and digital

Digital (89, 179, 65)-net over F₃, using

4 times m-reduction [i] based on digital (89, 183, 65)-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 (27, 121, 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]

(179−90, 179, 96)-Net over F₃ — Digital

Digital (89, 179, 96)-net over F₃, using

net from sequence [i] based on digital (89, 95)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 89 and N(F) ≥ 96, using
  - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(179−90, 179, 653)-Net in Base 3 — Upper bound on s

There is no (89, 179, 654)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 26 544776 604576 946204 548098 793767 699015 116949 071161 038954 251867 617139 167559 412772 619365 > 3¹⁷⁹ [i]