MinT - Best Known (6, 51, s)-Nets in Base 9

Best Known (6, 51, s)-Nets in Base 9

(6, 51, 34)-Net over F₉ — Constructive and digital

Digital (6, 51, 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, 51, 35)-Net over F₉ — Digital

Digital (6, 51, 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, 51, 85)-Net over F₉ — Upper bound on s (digital)

There is no digital (6, 51, 86)-net over F₉, because

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
      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]
  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, 51, 98)-Net in Base 9 — Upper bound on s

There is no (6, 51, 99)-net in base 9, because

extracting embedded orthogonal array [i] would yield OA(9⁵¹, 99, S₉, 45), but
- the linear programming bound shows that MÂ â‰¥ 1026 395312 362674 564882 877013 539504 511925 066027 347503 918341 088052 832186 909520 904385 108945 / 216 144366 363147 614695 574651 637919 756252 > 9⁵¹ [i]