MinT - Best Known (127−79, 127, s)-Nets in Base 3

Best Known (127−79, 127, s)-Nets in Base 3

(127−79, 127, 48)-Net over F₃ — Constructive and digital

Digital (48, 127, 48)-net over F₃, using

t-expansion [i] based on digital (45, 127, 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]

(127−79, 127, 56)-Net over F₃ — Digital

Digital (48, 127, 56)-net over F₃, using

t-expansion [i] based on digital (40, 127, 56)-net over F₃, using
- net from sequence [i] based on digital (40, 55)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 40 and N(F) ≥ 56, using
    - a curve from the manYPoints database [i]

(127−79, 127, 231)-Net in Base 3 — Upper bound on s

There is no (48, 127, 232)-net in base 3, because

1 times m-reduction [i] would yield (48, 126, 232)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 414510 743298 882523 970040 093663 085346 431294 994660 338749 905377 > 3¹²⁶ [i]