MinT - Best Known (30−20, 30, s)-Nets in Base 3

Best Known (30−20, 30, s)-Nets in Base 3

(30−20, 30, 19)-Net over F₃ — Constructive and digital

Digital (10, 30, 19)-net over F₃, using

t-expansion [i] based on digital (9, 30, 19)-net over F₃, using
- net from sequence [i] based on digital (9, 18)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 9 and N(F) ≥ 19, using
    - an explicitly constructive algebraic function field [i]

(30−20, 30, 20)-Net over F₃ — Digital

Digital (10, 30, 20)-net over F₃, using

net from sequence [i] based on digital (10, 19)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 10 and N(F) ≥ 20, using
  - a curve from the manYPoints database [i]

(30−20, 30, 42)-Net over F₃ — Upper bound on s (digital)

There is no digital (10, 30, 43)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3³⁰, 43, F₃, 20) (dual of [43, 13, 21]-code), but
- constructionÂ Y1 [i] would yield
  1. linear OA(3²⁹, 35, F₃, 20) (dual of [35, 6, 21]-code), but
    - â€œBouâ€ bound on codes from Brouwerâ€™s database [i]
  2. OA(3¹³, 43, S₃, 8), but
    - discarding factors would yield OA(3¹³, 41, S₃, 8), but
      - the Rao or (dual) Hamming bound shows that MÂ â‰¥ 1 708963 > 3¹³ [i]

(30−20, 30, 50)-Net in Base 3 — Upper bound on s

There is no (10, 30, 51)-net in base 3, because

extracting embedded orthogonal array [i] would yield OA(3³⁰, 51, S₃, 20), but
- the linear programming bound shows that MÂ â‰¥ 2 907201 254143 171726 717132 854591 / 13662 469385 049199 > 3³⁰ [i]