MinT - Best Known (8, 8+17, s)-Nets in Base 3

Best Known (8, 8+17, s)-Nets in Base 3

(8, 8+17, 16)-Net over F₃ — Constructive and digital

Digital (8, 25, 16)-net over F₃, using

t-expansion [i] based on digital (7, 25, 16)-net over F₃, using
- net from sequence [i] based on digital (7, 15)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 7 and N(F) ≥ 16, using
    - an explicitly constructive algebraic function field [i]

(8, 8+17, 17)-Net over F₃ — Digital

Digital (8, 25, 17)-net over F₃, using

net from sequence [i] based on digital (8, 16)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 8 and N(F) ≥ 17, using
  - a curve from the manYPoints database [i]

(8, 8+17, 36)-Net over F₃ — Upper bound on s (digital)

There is no digital (8, 25, 37)-net over F₃, because

2 times m-reduction [i] would yield digital (8, 23, 37)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²³, 37, F₃, 15) (dual of [37, 14, 16]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(3²², 28, F₃, 15) (dual of [28, 6, 16]-code), but
      - â€œHHMâ€ bound on codes from Brouwerâ€™s database [i]
    2. linear OA(3¹⁴, 37, F₃, 9) (dual of [37, 23, 10]-code), but
      - discarding factors / shortening the dual code would yield linear OA(3¹⁴, 31, F₃, 9) (dual of [31, 17, 10]-code), but
        residual code [i] would yield linear OA(3⁵, 21, F₃, 3) (dual of [21, 16, 4]-code or 21-cap in PG(4,3)), but
        20 is the largest size of a cap in PG(4,Â 3) [i]

(8, 8+17, 42)-Net in Base 3 — Upper bound on s

There is no (8, 25, 43)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3²⁵, 43, S₃, 17), but
- the linear programming bound shows that MÂ â‰¥ 1 258264 835074 776640 892553 / 1 376226 982273 > 3²⁵ [i]