MinT - Best Known (210−162, 210, s)-Nets in Base 3

Best Known (210−162, 210, s)-Nets in Base 3

(210−162, 210, 48)-Net over F₃ — Constructive and digital

Digital (48, 210, 48)-net over F₃, using

t-expansion [i] based on digital (45, 210, 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]

(210−162, 210, 56)-Net over F₃ — Digital

Digital (48, 210, 56)-net over F₃, using

t-expansion [i] based on digital (40, 210, 56)-net over F₃, using
- net from sequence [i] based on digital (40, 55)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 40 and N(F) ≥ 56, using
    - a curve from the manYPoints database [i]

(210−162, 210, 153)-Net over F₃ — Upper bound on s (digital)

There is no digital (48, 210, 154)-net over F₃, because

63 times m-reduction [i] would yield digital (48, 147, 154)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3¹⁴⁷, 154, F₃, 99) (dual of [154, 7, 100]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(3¹⁴⁶, 151, F₃, 99) (dual of [151, 5, 100]-code), but
      - Griesmer bound [i]
    2. linear OA(3⁷, 154, F₃, 3) (dual of [154, 147, 4]-code or 154-cap in PG(6,3)), but
      - discarding factors / shortening the dual code would yield linear OA(3⁷, 137, F₃, 3) (dual of [137, 130, 4]-code or 137-cap in PG(6,3)), but
        a result by BarÃ¡t, Edel, Hill, and Storme [i]

(210−162, 210, 156)-Net in Base 3 — Upper bound on s

There is no (48, 210, 157)-net in base 3, because

57 times m-reduction [i] would yield (48, 153, 157)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹⁵³, 157, S₃, 105), but
  - the (dual) Plotkin bound shows that MÂ â‰¥ 539 443211 181215 126524 213740 814573 707223 877780 368216 095822 538862 929948 947042 / 53 > 3¹⁵³ [i]