MinT - Best Known (108, 108+121, s)-Nets in Base 3

Best Known (108, 108+121, s)-Nets in Base 3

(108, 108+121, 74)-Net over F₃ — Constructive and digital

Digital (108, 229, 74)-net over F₃, using

t-expansion [i] based on digital (107, 229, 74)-net over F₃, using
- net from sequence [i] based on digital (107, 73)-sequence over F₃, using
  - base reduction for sequences [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]

(108, 108+121, 104)-Net over F₃ — Digital

Digital (108, 229, 104)-net over F₃, using

t-expansion [i] based on digital (102, 229, 104)-net over F₃, using
- net from sequence [i] based on digital (102, 103)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 102 and N(F) ≥ 104, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(108, 108+121, 696)-Net in Base 3 — Upper bound on s

There is no (108, 229, 697)-net in base 3, because

1 times m-reduction [i] would yield (108, 228, 697)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 6 390323 652869 145059 508712 854567 712672 440064 089895 157464 944681 667606 336071 414830 438765 357169 814959 397715 657457 > 3²²⁸ [i]