MinT - Best Known (54−29, 54, s)-Nets in Base 81

Best Known (54−29, 54, s)-Nets in Base 81

(54−29, 54, 370)-Net over F₈₁ — Constructive and digital

Digital (25, 54, 370)-net over F₈₁, using

t-expansion [i] based on digital (16, 54, 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]

(54−29, 54, 691)-Net over F₈₁ — Digital

Digital (25, 54, 691)-net over F₈₁, using

Gilbertâ€“VarÅ¡amov bound [i]

(54−29, 54, 1268657)-Net in Base 81 — Upper bound on s

There is no (25, 54, 1268658)-net in base 81, because

1 times m-reduction [i] would yield (25, 53, 1268658)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 141158 595931 765850 883758 358069 180822 208543 152228 641097 579003 726556 974267 360352 555681 154073 536385 383361 > 81⁵³ [i]