MinT - Best Known (3, 3+25, s)-Nets in Base 9

Best Known (3, 3+25, s)-Nets in Base 9

(3, 3+25, 28)-Net over F₉ — Constructive and digital

Digital (3, 28, 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, 3+25, 57)-Net over F₉ — Upper bound on s (digital)

There is no digital (3, 28, 58)-net over F₉, because

extracting embedded orthogonal array [i] would yield linear OA(9²⁸, 58, F₉, 25) (dual of [58, 30, 26]-code), but
- constructionÂ Y1 [i] would yield
  1. OA(9²⁷, 31, S₉, 25), but
    - the linear programming bound shows that MÂ â‰¥ 25644 034018 340666 323462 064529 / 377 > 9²⁷ [i]
  2. linear OA(9³⁰, 58, F₉, 27) (dual of [58, 28, 28]-code), but
    - discarding factors / shortening the dual code would yield linear OA(9³⁰, 39, F₉, 27) (dual of [39, 9, 28]-code), but
      - residual code [i] would yield OA(9³, 11, S₉, 3), but
        bound for OAs with index unity [i]

(3, 3+25, 73)-Net in Base 9 — Upper bound on s

There is no (3, 28, 74)-net in base 9, because

extracting embedded orthogonal array [i] would yield OA(9²⁸, 74, S₉, 25), but
- the linear programming bound shows that MÂ â‰¥ 72 750603 826730 008072 451570 559538 425525 / 131031 484853 > 9²⁸ [i]