MinT - Best Known (219−126, 219, s)-Nets in Base 3

Best Known (219−126, 219, s)-Nets in Base 3

(219−126, 219, 64)-Net over F₃ — Constructive and digital

Digital (93, 219, 64)-net over F₃, using

t-expansion [i] based on digital (89, 219, 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]

(219−126, 219, 96)-Net over F₃ — Digital

Digital (93, 219, 96)-net over F₃, using

t-expansion [i] based on digital (89, 219, 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]

(219−126, 219, 493)-Net in Base 3 — Upper bound on s

There is no (93, 219, 494)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 312 956955 222253 304337 993133 299508 615699 470296 466826 140003 109195 155636 121483 669624 069112 450548 054683 754185 > 3²¹⁹ [i]