MinT - Best Known (42−38, 42, s)-Nets in Base 9

Best Known (42−38, 42, s)-Nets in Base 9

(42−38, 42, 30)-Net over F₉ — Constructive and digital

Digital (4, 42, 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]

(42−38, 42, 44)-Net over F₉ — Upper bound on s (digital)

There is no digital (4, 42, 45)-net over F₉, because

2 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]

(42−38, 42, 50)-Net in Base 9 — Upper bound on s

There is no (4, 42, 51)-net in base 9, because

extracting embedded orthogonal array [i] would yield OA(9⁴², 51, S₉, 38), but
- the linear programming bound shows that MÂ â‰¥ 3 339519 604312 424791 531844 177789 823845 028679 340727 / 232 647415 > 9⁴² [i]