MinT - Best Known (222−156, 222, s)-Nets in Base 3

Best Known (222−156, 222, s)-Nets in Base 3

(222−156, 222, 48)-Net over F₃ — Constructive and digital

Digital (66, 222, 48)-net over F₃, using

t-expansion [i] based on digital (45, 222, 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]

(222−156, 222, 64)-Net over F₃ — Digital

Digital (66, 222, 64)-net over F₃, using

t-expansion [i] based on digital (49, 222, 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]

(222−156, 222, 206)-Net over F₃ — Upper bound on s (digital)

There is no digital (66, 222, 207)-net over F₃, because

21 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]

(222−156, 222, 211)-Net in Base 3 — Upper bound on s

There is no (66, 222, 212)-net in base 3, because

15 times m-reduction [i] would yield (66, 207, 212)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3²⁰⁷, 212, S₃, 141), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 47052 721287 394587 764057 094854 672253 553918 218437 190874 778408 030747 195017 485692 977810 906266 281547 645147 / 71 > 3²⁰⁷ [i]