MinT - Best Known (56−41, 56, s)-Nets in Base 3

Best Known (56−41, 56, s)-Nets in Base 3

(56−41, 56, 28)-Net over F₃ — Constructive and digital

Digital (15, 56, 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]

(56−41, 56, 53)-Net over F₃ — Upper bound on s (digital)

There is no digital (15, 56, 54)-net over F₃, because

5 times m-reduction [i] would yield digital (15, 51, 54)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3⁵¹, 54, F₃, 36) (dual of [54, 3, 37]-code), but
  - Griesmer bound [i]

(56−41, 56, 54)-Net in Base 3 — Upper bound on s

There is no (15, 56, 55)-net in base 3, because

8 times m-reduction [i] would yield (15, 48, 55)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3⁴⁸, 55, S₃, 33), but
  - the linear programming bound shows that MÂ â‰¥ 58 149737 003040 059690 390169 / 697 > 3⁴⁸ [i]