MinT - Best Known (136−85, 136, s)-Nets in Base 3

Best Known (136−85, 136, s)-Nets in Base 3

(136−85, 136, 48)-Net over F₃ — Constructive and digital

Digital (51, 136, 48)-net over F₃, using

t-expansion [i] based on digital (45, 136, 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]

(136−85, 136, 64)-Net over F₃ — Digital

Digital (51, 136, 64)-net over F₃, using

t-expansion [i] based on digital (49, 136, 64)-net over F₃, using
- net from sequence [i] based on digital (49, 63)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 49 and N(F) ≥ 64, using
    - a curve from the manYPoints database [i]

(136−85, 136, 242)-Net in Base 3 — Upper bound on s

There is no (51, 136, 243)-net in base 3, because

1 times m-reduction [i] would yield (51, 135, 243)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 26009 100228 726283 152079 529226 109728 682311 293688 686335 553647 946485 > 3¹³⁵ [i]