MinT - Best Known (222−71, 222, s)-Nets in Base 3

Best Known (222−71, 222, s)-Nets in Base 3

(222−71, 222, 168)-Net over F₃ — Constructive and digital

Digital (151, 222, 168)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (11, 46, 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]
2. digital (105, 176, 148)-net over F₃, using
  - trace code for nets [i] based on digital (17, 88, 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]

(222−71, 222, 397)-Net over F₃ — Digital

Digital (151, 222, 397)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(222−71, 222, 7125)-Net in Base 3 — Upper bound on s

There is no (151, 222, 7126)-net in base 3, because

1 times m-reduction [i] would yield (151, 221, 7126)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2784 116507 892496 355426 205721 337186 973381 124483 016513 252114 064987 057434 778510 160165 540927 774374 000372 995937 > 3²²¹ [i]