MinT - Best Known (66, 66+132, s)-Nets in Base 3

Best Known (66, 66+132, s)-Nets in Base 3

(66, 66+132, 48)-Net over F₃ — Constructive and digital

Digital (66, 198, 48)-net over F₃, using

t-expansion [i] based on digital (45, 198, 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]

(66, 66+132, 64)-Net over F₃ — Digital

Digital (66, 198, 64)-net over F₃, using

t-expansion [i] based on digital (49, 198, 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]

(66, 66+132, 209)-Net over F₃ — Upper bound on s (digital)

There is no digital (66, 198, 210)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3¹⁹⁸, 210, F₃, 132) (dual of [210, 12, 133]-code), but
- residual code [i] would yield OA(3⁶⁶, 77, S₃, 44), but
  - the linear programming bound shows that MÂ â‰¥ 34 245670 463812 538347 598340 341640 500169 / 1 016275 > 3⁶⁶ [i]

(66, 66+132, 282)-Net in Base 3 — Upper bound on s

There is no (66, 198, 283)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 33641 442847 409368 383965 750140 250127 575656 281074 271874 721718 513643 577057 757790 998110 173173 429717 > 3¹⁹⁸ [i]