MinT - Best Known (81−49, 81, s)-Nets in Base 81

Best Known (81−49, 81, s)-Nets in Base 81

(81−49, 81, 370)-Net over F₈₁ — Constructive and digital

Digital (32, 81, 370)-net over F₈₁, using

t-expansion [i] based on digital (16, 81, 370)-net over F₈₁, using
- net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
    - a function field by GarcÃa and Quoos [i]

(81−49, 81, 551)-Net over F₈₁ — Digital

Digital (32, 81, 551)-net over F₈₁, using

t-expansion [i] based on digital (30, 81, 551)-net over F₈₁, using
- net from sequence [i] based on digital (30, 550)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 30 and N(F) ≥ 551, using
    - a curve from the manYPoints database [i]

(81−49, 81, 281754)-Net in Base 81 — Upper bound on s

There is no (32, 81, 281755)-net in base 81, because

1 times m-reduction [i] would yield (32, 80, 281755)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 477 320062 070673 608172 820724 400784 345577 217356 209860 719844 696113 944533 006867 057170 719934 423502 255540 330282 162900 703589 512993 340321 108201 774589 343317 545601 > 81⁸⁰ [i]