MinT - Best Known (6, 6+47, s)-Nets in Base 9

Best Known (6, 6+47, s)-Nets in Base 9

(6, 6+47, 34)-Net over F₉ — Constructive and digital

Digital (6, 53, 34)-net over F₉, using

net from sequence [i] based on digital (6, 33)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 6 and N(F) ≥ 34, using
  - a function field by SÃ©mirat [i]

(6, 6+47, 35)-Net over F₉ — Digital

Digital (6, 53, 35)-net over F₉, using

net from sequence [i] based on digital (6, 34)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 6 and N(F) ≥ 35, using
  - Drinfeld modules of rank 1 [i]

(6, 6+47, 85)-Net over F₉ — Upper bound on s (digital)

There is no digital (6, 53, 86)-net over F₉, because

2 times m-reduction [i] would yield digital (6, 51, 86)-net over F₉, but
- extracting embedded orthogonal array [i] would yield linear OA(9⁵¹, 86, F₉, 45) (dual of [86, 35, 46]-code), but
  - constructionÂ Y1 [i] would yield
    1. linear OA(9⁵⁰, 55, F₉, 45) (dual of [55, 5, 46]-code), but
      - constructionÂ Y1 [i] would yield
        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]
        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]
    2. OA(9³⁵, 86, S₉, 31), but
      - discarding factors would yield OA(9³⁵, 85, S₉, 31), but
        the linear programming bound shows that MÂ â‰¥ 112620 123087 737061 231244 499443 215625 747636 066957 184847 246285 / 44 721454 781644 057928 595589 > 9³⁵ [i]

(6, 6+47, 90)-Net in Base 9 — Upper bound on s

There is no (6, 53, 91)-net in base 9, because

extracting embedded orthogonal array [i] would yield OA(9⁵³, 91, S₉, 47), but
- the linear programming bound shows that MÂ â‰¥ 1 805866 253412 423561 017231 691027 796883 544324 423036 889068 209415 907113 599556 237288 179883 / 4654 018646 492649 551136 876628 773107 > 9⁵³ [i]