MinT - Best Known (66, 205, s)-Nets in Base 3

Best Known (66, 205, s)-Nets in Base 3

(66, 205, 48)-Net over F₃ — Constructive and digital

Digital (66, 205, 48)-net over F₃, using

t-expansion [i] based on digital (45, 205, 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, 205, 64)-Net over F₃ — Digital

Digital (66, 205, 64)-net over F₃, using

t-expansion [i] based on digital (49, 205, 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, 205, 206)-Net over F₃ — Upper bound on s (digital)

There is no digital (66, 205, 207)-net over F₃, because

4 times m-reduction [i] would yield digital (66, 201, 207)-net over F₃, but
- 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, 205, 278)-Net in Base 3 — Upper bound on s

There is no (66, 205, 279)-net in base 3, because

1 times m-reduction [i] would yield (66, 204, 279)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 26 037292 023120 767771 615270 610401 068594 933396 235487 816665 746036 479901 555959 081824 946347 857059 630951 > 3²⁰⁴ [i]