MinT - Best Known (159−89, 159, s)-Nets in Base 3

Best Known (159−89, 159, s)-Nets in Base 3

(159−89, 159, 48)-Net over F₃ — Constructive and digital

Digital (70, 159, 48)-net over F₃, using

t-expansion [i] based on digital (45, 159, 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]

(159−89, 159, 82)-Net over F₃ — Digital

Digital (70, 159, 82)-net over F₃, using

t-expansion [i] based on digital (69, 159, 82)-net over F₃, using
- net from sequence [i] based on digital (69, 81)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 69 and N(F) ≥ 82, using
    - Drinfeld modules of rank 1 [i]

(159−89, 159, 404)-Net in Base 3 — Upper bound on s

There is no (70, 159, 405)-net in base 3, because

1 times m-reduction [i] would yield (70, 158, 405)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2670 950460 758984 764579 740145 588770 267442 808372 558396 004737 693904 219629 799985 > 3¹⁵⁸ [i]