MinT - Best Known (3, 57, s)-Nets in Base 9

Best Known (3, 57, s)-Nets in Base 9

(3, 57, 28)-Net over F₉ — Constructive and digital

Digital (3, 57, 28)-net over F₉, using

net from sequence [i] based on digital (3, 27)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 3 and N(F) ≥ 28, using
  - the Hermitian function field over F₉ [i]

(3, 57, 36)-Net over F₉ — Upper bound on s (digital)

There is no digital (3, 57, 37)-net over F₉, because

27 times m-reduction [i] would yield digital (3, 30, 37)-net over F₉, but
- extracting embedded orthogonal array [i] would yield linear OA(9³⁰, 37, F₉, 27) (dual of [37, 7, 28]-code), but
  - constructionÂ Y1 [i] would yield
    1. OA(9²⁹, 31, S₉, 27), but
      - the (dual) Plotkin bound shows that MÂ â‰¥ 42391 158275 216203 514294 433201 / 7 > 9²⁹ [i]
    2. OA(9⁷, 37, S₉, 6), but
      - the linear programming bound shows that MÂ â‰¥ 8306 954271 / 1613 > 9⁷ [i]

(3, 57, 40)-Net in Base 9 — Upper bound on s

There is no (3, 57, 41)-net in base 9, because

20 times m-reduction [i] would yield (3, 37, 41)-net in base 9, but
- extracting embedded orthogonal array [i] would yield OA(9³⁷, 41, S₉, 34), but
  - the linear programming bound shows that MÂ â‰¥ 494 520898 410959 817514 303085 999134 242591 / 2275 > 9³⁷ [i]