MinT - Best Known (222−55, 222, s)-Nets in Base 3

Best Known (222−55, 222, s)-Nets in Base 3

(222−55, 222, 328)-Net over F₃ — Constructive and digital

Digital (167, 222, 328)-net over F₃, using

3² times duplication [i] based on digital (165, 220, 328)-net over F₃, using
- trace code for nets [i] based on digital (0, 55, 82)-net over F₈₁, using
  - net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
      - the rational function field F₈₁(x) [i]
    - Niederreiter sequence [i]

(222−55, 222, 959)-Net over F₃ — Digital

Digital (167, 222, 959)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(222−55, 222, 43897)-Net in Base 3 — Upper bound on s

There is no (167, 222, 43898)-net in base 3, because

1 times m-reduction [i] would yield (167, 221, 43898)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2779 134880 852798 953200 407500 817692 843529 187506 838817 436395 808590 726925 234569 310191 856846 181312 551189 605121 > 3²²¹ [i]