MinT - Best Known (82−77, 82, s)-Nets in Base 9

Best Known (82−77, 82, s)-Nets in Base 9

(82−77, 82, 32)-Net over F₉ — Constructive and digital

Digital (5, 82, 32)-net over F₉, using

net from sequence [i] based on digital (5, 31)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 5 and N(F) ≥ 32, using
  - a function field by GarcÃa and Quoos [i]

(82−77, 82, 54)-Net over F₉ — Upper bound on s (digital)

There is no digital (5, 82, 55)-net over F₉, because

32 times m-reduction [i] would yield digital (5, 50, 55)-net over F₉, but
- extracting embedded orthogonal array [i] would yield linear OA(9⁵⁰, 55, F₉, 45) (dual of [55, 5, 46]-code), but
  - constructionÂ Y1 [i] would yield
    1. OA(9⁴⁹, 51, S₉, 45), but
      - the (dual) Plotkin bound shows that MÂ â‰¥ 1 546132 562196 033993 109383 389296 863818 106322 566003 / 23 > 9⁴⁹ [i]
    2. OA(9⁵, 55, 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]

(82−77, 82, 56)-Net in Base 9 — Upper bound on s

There is no (5, 82, 57)-net in base 9, because

32 times m-reduction [i] would yield (5, 50, 57)-net in base 9, but
- extracting embedded orthogonal array [i] would yield OA(9⁵⁰, 57, S₉, 45), but
  - the linear programming bound shows that MÂ â‰¥ 298105 180919 330726 109440 883140 160422 177625 734983 340421 / 537625 > 9⁵⁰ [i]