MinT - Best Known (66, 201, s)-Nets in Base 3

Best Known (66, 201, s)-Nets in Base 3

(66, 201, 48)-Net over F₃ — Constructive and digital

Digital (66, 201, 48)-net over F₃, using

t-expansion [i] based on digital (45, 201, 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, 201, 64)-Net over F₃ — Digital

Digital (66, 201, 64)-net over F₃, using

t-expansion [i] based on digital (49, 201, 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, 201, 206)-Net over F₃ — Upper bound on s (digital)

There is no digital (66, 201, 207)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²⁰¹, 207, F₃, 135) (dual of [207, 6, 136]-code), but
- residual code [i] would yield linear OA(3⁶⁶, 71, F₃, 45) (dual of [71, 5, 46]-code), but
  - â€œHW1â€ bound on codes from Brouwerâ€™s database [i]

(66, 201, 280)-Net in Base 3 — Upper bound on s

There is no (66, 201, 281)-net in base 3, because

1 times m-reduction [i] would yield (66, 200, 281)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 275640 491943 162744 714236 352677 857889 580333 578854 420588 671642 898395 558397 014566 560418 396536 000059 > 3²⁰⁰ [i]