MinT - Best Known (65, 203, s)-Nets in Base 3

Best Known (65, 203, s)-Nets in Base 3

(65, 203, 48)-Net over F₃ — Constructive and digital

Digital (65, 203, 48)-net over F₃, using

t-expansion [i] based on digital (45, 203, 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]

(65, 203, 64)-Net over F₃ — Digital

Digital (65, 203, 64)-net over F₃, using

t-expansion [i] based on digital (49, 203, 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]

(65, 203, 204)-Net over F₃ — Upper bound on s (digital)

There is no digital (65, 203, 205)-net over F₃, because

3 times m-reduction [i] would yield digital (65, 200, 205)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²⁰⁰, 205, F₃, 135) (dual of [205, 5, 136]-code), but
  - Griesmer bound [i]

(65, 203, 272)-Net in Base 3 — Upper bound on s

There is no (65, 203, 273)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7 554419 427267 052709 643445 040207 196908 958408 525233 695540 439071 289764 889709 156098 969868 340338 463251 > 3²⁰³ [i]