MinT - Best Known (50−34, 50, s)-Nets in Base 3

Best Known (50−34, 50, s)-Nets in Base 3

(50−34, 50, 28)-Net over F₃ — Constructive and digital

Digital (16, 50, 28)-net over F₃, using

t-expansion [i] based on digital (15, 50, 28)-net over F₃, using
- net from sequence [i] based on digital (15, 27)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
    - an explicitly constructive algebraic function field [i]

(50−34, 50, 57)-Net over F₃ — Upper bound on s (digital)

There is no digital (16, 50, 58)-net over F₃, because

1 times m-reduction [i] would yield digital (16, 49, 58)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3⁴⁹, 58, F₃, 33) (dual of [58, 9, 34]-code), but
  - residual code [i] would yield linear OA(3¹⁶, 24, F₃, 11) (dual of [24, 8, 12]-code), but
    - â€œBSâ€ bound on codes from Brouwerâ€™s database [i]

(50−34, 50, 59)-Net in Base 3 — Upper bound on s

There is no (16, 50, 60)-net in base 3, because

1 times m-reduction [i] would yield (16, 49, 60)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3⁴⁹, 60, S₃, 33), but
  - the linear programming bound shows that MÂ â‰¥ 224516 134568 737670 464596 442509 / 753457 > 3⁴⁹ [i]