MinT - Best Known (239−76, 239, s)-Nets in Base 3

Best Known (239−76, 239, s)-Nets in Base 3

(239−76, 239, 176)-Net over F₃ — Constructive and digital

Digital (163, 239, 176)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 53, 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 (110, 186, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 93, 74)-net over F₉, using
    - net from sequence [i] based on digital (17, 73)-sequence over F₉, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
        a function field by GarcÃa and Quoos [i]

(239−76, 239, 431)-Net over F₃ — Digital

Digital (163, 239, 431)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(239−76, 239, 7489)-Net in Base 3 — Upper bound on s

There is no (163, 239, 7490)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 079388 572528 603163 375698 739166 238432 222577 854681 469303 590705 392326 174134 818968 470619 924810 019376 172227 715843 859197 > 3²³⁹ [i]