MinT - Best Known (4, 68, s)-Nets in Base 9

Best Known (4, 68, s)-Nets in Base 9

(4, 68, 30)-Net over F₉ — Constructive and digital

Digital (4, 68, 30)-net over F₉, using

net from sequence [i] based on digital (4, 29)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 4 and N(F) ≥ 30, using
  - a function field by SÃ©mirat [i]

(4, 68, 44)-Net over F₉ — Upper bound on s (digital)

There is no digital (4, 68, 45)-net over F₉, because

28 times m-reduction [i] would yield digital (4, 40, 45)-net over F₉, but
- extracting embedded orthogonal array [i] would yield linear OA(9⁴⁰, 45, F₉, 36) (dual of [45, 5, 37]-code), but
  - constructionÂ Y1 [i] would yield
    1. OA(9³⁹, 41, S₉, 36), but
      - the (dual) Plotkin bound shows that MÂ â‰¥ 739 044147 071729 616580 416051 031916 488005 / 37 > 9³⁹ [i]
    2. OA(9⁵, 45, S₉, 4), but
      - discarding factors would yield OA(9⁵, 44, S₉, 4), but
        the Rao or (dual) Hamming bound shows that MÂ â‰¥ 60897 > 9⁵ [i]

(4, 68, 49)-Net in Base 9 — Upper bound on s

There is no (4, 68, 50)-net in base 9, because

25 times m-reduction [i] would yield (4, 43, 50)-net in base 9, but
- extracting embedded orthogonal array [i] would yield OA(9⁴³, 50, S₉, 39), but
  - the linear programming bound shows that MÂ â‰¥ 307 418272 342644 982111 952555 666013 618532 571037 / 2068 > 9⁴³ [i]