MinT - Best Known (51, 126, s)-Nets in Base 9

Best Known (51, 126, s)-Nets in Base 9

(51, 126, 81)-Net over F₉ — Constructive and digital

Digital (51, 126, 81)-net over F₉, using

t-expansion [i] based on digital (32, 126, 81)-net over F₉, using
- net from sequence [i] based on digital (32, 80)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
    - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(51, 126, 88)-Net in Base 9 — Constructive

(51, 126, 88)-net in base 9, using

base change [i] based on digital (9, 84, 88)-net over F₂₇, using
- net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
    - a function field by SÃ©mirat [i]

(51, 126, 182)-Net over F₉ — Digital

Digital (51, 126, 182)-net over F₉, using

t-expansion [i] based on digital (50, 126, 182)-net over F₉, using
- net from sequence [i] based on digital (50, 181)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 50 and N(F) ≥ 182, using
    - a curve from the manYPoints database [i]

(51, 126, 3043)-Net in Base 9 — Upper bound on s

There is no (51, 126, 3044)-net in base 9, because

1 times m-reduction [i] would yield (51, 125, 3044)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 191329 011661 821848 101927 570311 827796 675144 332781 266830 234828 068792 393291 905444 528394 463258 100336 616016 817819 617989 761185 > 9¹²⁵ [i]