MinT - Best Known (209−144, 209, s)-Nets in Base 3

Best Known (209−144, 209, s)-Nets in Base 3

(209−144, 209, 48)-Net over F₃ — Constructive and digital

Digital (65, 209, 48)-net over F₃, using

t-expansion [i] based on digital (45, 209, 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]

(209−144, 209, 64)-Net over F₃ — Digital

Digital (65, 209, 64)-net over F₃, using

t-expansion [i] based on digital (49, 209, 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]

(209−144, 209, 204)-Net over F₃ — Upper bound on s (digital)

There is no digital (65, 209, 205)-net over F₃, because

9 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]

(209−144, 209, 208)-Net in Base 3 — Upper bound on s

There is no (65, 209, 209)-net in base 3, because

5 times m-reduction [i] would yield (65, 204, 209)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3²⁰⁴, 209, S₃, 139), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 1742 693381 014614 361631 744253 876750 131626 600682 858921 288089 186323 970185 832803 443622 626158 010427 690561 / 70 > 3²⁰⁴ [i]