MinT - Best Known (171−99, 171, s)-Nets in Base 3

Best Known (171−99, 171, s)-Nets in Base 3

(171−99, 171, 48)-Net over F₃ — Constructive and digital

Digital (72, 171, 48)-net over F₃, using

t-expansion [i] based on digital (45, 171, 48)-net over F₃, using
- net from sequence [i] based on digital (45, 47)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
    - an explicitly constructive algebraic function field [i]

(171−99, 171, 84)-Net over F₃ — Digital

Digital (72, 171, 84)-net over F₃, using

t-expansion [i] based on digital (71, 171, 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]

(171−99, 171, 385)-Net in Base 3 — Upper bound on s

There is no (72, 171, 386)-net in base 3, because

1 times m-reduction [i] would yield (72, 170, 386)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1327 076057 150647 056258 465278 220633 359472 320941 091238 087475 390251 167833 929825 105957 > 3¹⁷⁰ [i]