MinT - Best Known (10, 10+19, s)-Nets in Base 3

Best Known (10, 10+19, s)-Nets in Base 3

(10, 10+19, 19)-Net over F₃ — Constructive and digital

Digital (10, 29, 19)-net over F₃, using

t-expansion [i] based on digital (9, 29, 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]

(10, 10+19, 20)-Net over F₃ — Digital

Digital (10, 29, 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]

(10, 10+19, 46)-Net over F₃ — Upper bound on s (digital)

There is no digital (10, 29, 47)-net over F₃, because

1 times m-reduction [i] would yield digital (10, 28, 47)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3²⁸, 47, F₃, 18) (dual of [47, 19, 19]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(3²⁷, 35, F₃, 18) (dual of [35, 8, 19]-code), but
      - residual code [i] would yield linear OA(3⁹, 16, F₃, 6) (dual of [16, 7, 7]-code), but
        â€œvE2â€ bound on codes from Brouwerâ€™s database [i]
    2. linear OA(3¹⁹, 47, F₃, 12) (dual of [47, 28, 13]-code), but
      - discarding factors / shortening the dual code would yield linear OA(3¹⁹, 46, F₃, 12) (dual of [46, 27, 13]-code), but
        residual code [i] would yield OA(3⁷, 33, S₃, 4), but
        the linear programming bound shows that MÂ â‰¥ 64125 / 28 > 3⁷ [i]

(10, 10+19, 55)-Net in Base 3 — Upper bound on s

There is no (10, 29, 56)-net in base 3, because

1 times m-reduction [i] would yield (10, 28, 56)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 25 611589 036785 > 3²⁸ [i]