MinT - Best Known (3, 29, s)-Nets in Base 9

Best Known (3, 29, s)-Nets in Base 9

(3, 29, 28)-Net over F₉ — Constructive and digital

Digital (3, 29, 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, 29, 57)-Net over F₉ — Upper bound on s (digital)

There is no digital (3, 29, 58)-net over F₉, because

1 times m-reduction [i] would yield digital (3, 28, 58)-net over F₉, but
- 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, 29, 69)-Net in Base 9 — Upper bound on s

There is no (3, 29, 70)-net in base 9, because

extracting embedded orthogonal array [i] would yield OA(9²⁹, 70, S₉, 26), but
- the linear programming bound shows that MÂ â‰¥ 142987 262234 073920 151202 698779 772123 777873 / 27 836552 119093 > 9²⁹ [i]