MinT - Best Known (170−91, 170, s)-Nets in Base 3

Best Known (170−91, 170, s)-Nets in Base 3

(170−91, 170, 56)-Net over F₃ — Constructive and digital

Digital (79, 170, 56)-net over F₃, using

7 times m-reduction [i] based on digital (79, 177, 56)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 64, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
        an explicitly constructive algebraic function field [i]
  2. digital (15, 113, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(170−91, 170, 84)-Net over F₃ — Digital

Digital (79, 170, 84)-net over F₃, using

t-expansion [i] based on digital (71, 170, 84)-net over F₃, using
- net from sequence [i] based on digital (71, 83)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 71 and N(F) ≥ 84, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(170−91, 170, 502)-Net in Base 3 — Upper bound on s

There is no (79, 170, 503)-net in base 3, because

1 times m-reduction [i] would yield (79, 169, 503)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 440 621269 545056 934525 030321 663393 800857 277247 780453 903750 619996 304138 912825 431383 > 3¹⁶⁹ [i]