MinT - Best Known (15, 47, s)-Nets in Base 3

Best Known (15, 47, s)-Nets in Base 3

(15, 47, 28)-Net over F₃ — Constructive and digital

Digital (15, 47, 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]

(15, 47, 54)-Net over F₃ — Upper bound on s (digital)

There is no digital (15, 47, 55)-net over F₃, because

1 times m-reduction [i] would yield digital (15, 46, 55)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3⁴⁶, 55, F₃, 31) (dual of [55, 9, 32]-code), but
  - â€œGurâ€ bound on codes from Brouwerâ€™s database [i]

(15, 47, 56)-Net in Base 3 — Upper bound on s

There is no (15, 47, 57)-net in base 3, because

1 times m-reduction [i] would yield (15, 46, 57)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3⁴⁶, 57, S₃, 31), but
  - the linear programming bound shows that MÂ â‰¥ 109 838392 116853 446081 848097 / 11440 > 3⁴⁶ [i]