MinT - Best Known (56−40, 56, s)-Nets in Base 3

Best Known (56−40, 56, s)-Nets in Base 3

(56−40, 56, 28)-Net over F₃ — Constructive and digital

Digital (16, 56, 28)-net over F₃, using

t-expansion [i] based on 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]

(56−40, 56, 55)-Net over F₃ — Upper bound on s (digital)

There is no digital (16, 56, 56)-net over F₃, because

4 times m-reduction [i] would yield digital (16, 52, 56)-net over F₃, but
- extracting embedded orthogonal array [i] would yield linear OA(3⁵², 56, F₃, 36) (dual of [56, 4, 37]-code), but
  - Griesmer bound [i]

(56−40, 56, 57)-Net in Base 3 — Upper bound on s

There is no (16, 56, 58)-net in base 3, because

5 times m-reduction [i] would yield (16, 51, 58)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3⁵¹, 58, S₃, 35), but
  - the linear programming bound shows that MÂ â‰¥ 2151 540269 112482 208544 436253 / 946 > 3⁵¹ [i]