MinT - Best Known (249−157, 249, s)-Nets in Base 3

Best Known (249−157, 249, s)-Nets in Base 3

(249−157, 249, 64)-Net over F₃ — Constructive and digital

Digital (92, 249, 64)-net over F₃, using

t-expansion [i] based on digital (89, 249, 64)-net over F₃, using
- net from sequence [i] based on digital (89, 63)-sequence over F₃, using
  - base reduction for sequences [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(249−157, 249, 96)-Net over F₃ — Digital

Digital (92, 249, 96)-net over F₃, using

t-expansion [i] based on digital (89, 249, 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]

(249−157, 249, 418)-Net in Base 3 — Upper bound on s

There is no (92, 249, 419)-net in base 3, because

1 times m-reduction [i] would yield (92, 248, 419)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 24366 782260 357738 226119 343775 574083 464166 960053 054706 904525 379528 714083 057075 690829 067956 717497 417656 376814 474880 975885 > 3²⁴⁸ [i]